Specifically, could new planets form around a previously destroyed star? (Think of it like a phoenix.) If so, could the potential sentient life detect that their star previously exploded?
If by explode you mean go Supernova, then no. Emphatically, no.
Most stars don't explode at their end of life...they turn into a Red Giant, then a White Dwarf, then they burn out. This is a one-way street.
There are, in general, two main ways that a star can go boom. Either you can have a Core Collapse (type 2), which is when a massive star's core destabilizes (for one of several reason), collapses--creating a massive energy surge--and it detonates. The other is when you have a pair of stars...one a white dwarf, the other large...and the white dwarf is stealing matter from the larger star (Type 1a). Eventually, it steals enough matter that it exceeds the Chandrasekhar limit, collapses, and then explodes. (there are one or two other ways it can blow, such as white dwarfs colliding...but they are rare occurrences)
When a star goes Supernova, the resulting explosion is the second most energetic type of event that we have ever recorded, exceeded only by colliding black holes, and the energy they released was mostly gravitational in nature. The luminosity of an exploding star can exceed the luminosity of a galaxy. Betelgeuse, a red giant star that is 640 light years away, is nearing the end of its life and scientists expect it will go out with a bang. When it does, the resulting flare of light as observed from Earth will be as bright, or brighter, than a full moon.
In a 'What-If' XKCD article, Randall Monroe spoke with a Physicist friend who said "however powerful you think a Supernova is...it's more powerful than that." If you compare a Hydrogen Bomb pressed against your eyeball and a Supernova viewed from the distance Earth is from the Sun...the Supernova will deliver more energy to you by 1,000,000,000 times (ref)
Finally, the surviving remnants after a Supernova depends on the type. A Type 1a supernova leaves nothing behind but a rapidly expanding 'supernova remnant'. There is no star left behind for anything to form around. A Type 2 will form into either a Neutron Star or a Black Hole as the super-dense core of the star is left behind after the explosion.
I'll get to why all of that is important in a second...
The next thing to examine is how planets form around a star. In a nebula, gasses can reach a point where there is enough in one place to start to generate a gravity field strong enough to pull more gas/dust towards itself. This creates a cascading process where it draws more and more gas in towards itself, becoming more and more massive as it goes. It also starts to spin as this happens. If enough gas/dust accrues, fusion ignites and a star is born. The remaining gas and dust are strung out in a disc around the newborn protostar, and...now that the new solar wind and radiation pressure is pushing the dust away from the star to counteract the gravity pulling it towards the star...the dust is now a little more likely to settle into an orbit, forming a protoplanetary disc. During this time, it doesn't stop clumping, and the resulting smaller bits that accrete together are how you get planets.
So, in order to get planets, you need that dust cloud for them to form from.
So, to finally give the answer, let's bring all of this together.
A Type 1 Supernova cannot meet your requirements: there's no star left for planets to form around. So that's out and only leaves a Type 2. A sufficiently massive Type 2 will create a Black Hole; those don't turn back into stars. A smaller one will create a Neutron star, but those don't turn back into normal stars either; if you add mass to a Neutron star, they collapse into a black hole. Black holes are obviously unfriendly to life, but neutron stars aren't much better. They don't emit a ton of energy, aren't very bright, and most of what they emit is in the form of blasts of radio waves.
Furthermore, the insane forces involved in a Supernova produce a shockwave that has been clocked at moving around 10% of the speed of light that slams into all matter within the stellar system and hurls it away from the point of the explosion. This creates what we call a Supernova Remnant, which is a rapidly expanding cloud of super-heated gas and debris. A prominent one that we know about is Cassiopeia A, which is (as viewed from Earth) about 10 light-years across and is still expanding at a velocity of 4-6,000 km/s.
The vast bulk of the matter in the stellar system will be ejected from the system. However, due to interactions between the matter, it is possible for some matter to not be ejected from the system. For clarity, this means that the matter ejected from the star with the explosion collides with planets, with other ejecta from the star moving at different speeds, and loses velocity. If it runs into enough other matter and loses enough speed, the gravity of the surviving core of the star (neutron star or black hole) overcomes its velocity and pulls the matter back in. Most of this matter will be pulled back into the remnants of the stellar core. However, it is possible for some of it to gain enough angular velocity to enter orbit around the stellar core (now a Neutron Star or Black Hole), creating a circumstellar disc. If there is enough mass in this disc, it is possible for planets to form out of it. Alternately, we have a few cases where it looks like the core of a companion star survived the Supernova and formed something vaguely planet-like.
However, this does not change the answer with regards to anything living there, because neutron stars and black holes are incredibly lethal to life as we know it, pouring massive doses of ionizing radiation into the space around them.
So, in summary: A Type 1a Supernova fails your requirements, because it leaves nothing star-like behind. A Type 2, at best, leaves behind a low-energy star that is very unfriendly to life, otherwise it leaves behind a black hole...and if material falls back into the low-energy star, you still get a black hole. In certain cases, planets may still form, but they will be uninhabitable unless you get into extremely exotic 'life' that doesn't operate anything like life as we currently know it.
Yes to the first, No to the second.
Planets may in fact be common around neutron stars. Debris from the supernova forms a circumstellar disk around the neutron star and, although composed of heavy metals rather than light dust, is similar to the disk where planets formed around a new star so similar dynamics are presumed.
Watching the process of how these planets form so quickly after the supernova is cutting edge astronomy.
However, it's been known for 25 years that pulsar planets exist. While one such body appears to have been capured from its companion star, in general they appear to have formed in place after the supernova.
As for life, life as we know it could not exist under these conditions. Some exotic life like hard s-f authors dream up would be interesting to discuss on its own Question if it could be asked in a way that’s not “too broad” or “opinion based”.
A star that explodes sufficiently violently to destroy itself is called a Supernova, and they come in several types. Some kinds of supernova leave a remnant, in the form of a neutron star or a black hole.
Firstly, a supernova explosion is so violent that all the matter blown away by the explosion will be moving above any remnant's escape velocity, so there's no possibility of the star reforming: that can't happen.
It is possible that a black hole or neutron star could have planets remaining from before the supernova. Or rather, the remnants of planets. To get an idea of what a supernova will do to its planets, make a model planet out of butter or jello, and then apply a welding torch, enthusiastically and for several minutes. You might have a small crisped body left, if you started with a large one, but all life on it will certainly have died, and the planet will be much smaller than it used to be.
It is just conceivably possible that new life could arise on such a remnant planet, although it's deeply implausible: the environment will be very hostile. If such life developed sentience and science, they might eventually figure out that their "star" is quite unusual and had exploded far in the past.
However, there is a much more interesting answer. The Sun, Earth, and the entire solar system are made out of debris from one or more supernova explosions. The superheated gas blown off into space from a supernova can eventually form new stars, but note that this is likely to be several small stars, and there's no real way in which they are the original star. We know that we're made out of the debris of a supernova (or maybe more than one) because Earth has heavy elements like gold and uranium. A supernova explosion is the only process in the universe that produces such heavy elements, as far as we know.
So you, and everything around you, are made of matter that has been part of a different star, which exploded. Which is rather cool.
There is no simple answer to your question. In a way, yes, exploding stars can (and do) create planets. And yes, sentient life can (and does, in our case) understand that their star exploded in the past. Yet, what guildsbounty has stated above is pragmatically correct. Let me explain.
There have been three generations of stars. The first generation stars (also known as population III stars) were stars which were monstrously huge, even when compared to our sun (and our sun is a seriously gigantic object). They all exploded in supernovas known as pair instability supernova. The flesh and bone (more like debris) of these dead stars later condensed to form second generation stars which were relatively rich in heavy elements (anything with an atomic number greater than 2 is known as heavy element in cosmology). Then these second generation stars died too, and their remains aggregated and produced third generation stars, one of which is our sun.
So yes, we are literally stars. Our Earth contains large amounts of metals, carbon, oxygen, silicon, halogens and several other elements, all of which were formed in the core of a huge star, once. And this earthly dust forms our bodies, our flesh and bones, skin and blood. And we, being sentient species, know that we (and our Earth, and our whole solar system) is here as a result of a horrific stellar explosion billions of years ago.
So yes, stars do explode and their contents later condense together to form newer, smaller stars, and planets. But so far as we know about supernovas, no planetary system is known to survive the violent explosion of its parent star. The shockwave resulting from the supernova destroys planets within minutes, leaving only dust and space debris. Later this dust and debris resettles to form newer solar systems, but the older, original planets do not remain and a star's remains do not add new planets to the older planetary system. Once a supernova occurs, the whole system is destroyed and rebuilt in the next few billions of years.
There is a caveat most people have not considered - what becomes of the stellar dust which was your star after the supernova scatters it to the galactic winds.
Flung out into the interstellar medium, riding an immense and highly energetic shockwave, if this matter encounters nebular matter, it is likely to accrete and form new stellar masses, which in turn may form perfectly normal planetary systems. Indeed, most of the stars you see in the sky were formed this way; clumps of stuff smashed together by the shockwave of a dying ancestor. Part of their mass is composed of the baryons from the dead star, thus semi-meeting the requirement of being the dead star.
Could this be detected? Yes, probably, depending on the structure of the interstellar neighborhood in which these inhabitants found themselves. If you look out at some of the nebula in our own galaxy, you can find clusters of similarly-aged stars which appear to ring an ancient supernova site. Such a structure could also be detected from within itself by one of those system's inhabitants.
One supernova probably won't be enough to make a new star but couple of them will make couple of stars. And for a planet to be habitable, it must be formed in such a situation. Our star is at least third generation star. So your idea of Phoenix star system won't happen, but a Phoenix cluster might work.
This is basically the same answer as Youstay Igo, but simpler.