How would an organism evolve to ''see''or sense things with high precision way before light hits its eyes or sound reaches their ears, or at least giving the illusion of doing so?
If you want the creature to actually be able to sense before the light or soundwaves reach it, you're in trouble. There's some severe causality challenges which come with sensing something before it can be sensed.
In particular, if you want it to see something before light hits its eyes, you literally have to break the laws of physics: relativity and Maxwell's equations in particular. Maxwell's equations specify the speed of light, and relativity shows that we cannot observe any signals faster than light. It's a law of physics, and has so much evidence behind it that you're basically on your own here, making up a new world.
You could play games with an organism that can generate wormholes, giving it a way to shortcut much of that distance out. However, this doesn't quite fit what you need. All you did was give light a shorter path to take; the light still had to reach its eyes before being detected. Even then, the mathematics of such wormholes put some rather significant limits on where this could go before you have to rewrite the laws of physics.
Hearing something before the sound waves reach the ears is more doable, because sound waves travel so much slower. We have a device called a laser microphone which shines a laser on a surface far away to detect vibrations. Because the laser travels so much faster than sound waves, this technique can be used to be aware of sound vibrations from far away long before the sound vibrations actually reach you. Evolving such a system would be a bit of a feat, though. These systems tend to be quite precise. There are lasers which rely on organic dyes for lasing though, so the chemistry is theoretically reasonable.
A more interesting approach would be giving the illusion of being able to perceive at a distance. To do this, one of these organisms would need to understand the system which would be emitting light or sound better than you understand it, and be able to sense the precursors to that event. A very clearcut example can be in war with grenades. If someone tosses a grenade into a room, you have a few seconds before it emits light and energy... a lot of light and energy (and messy metal bits). If you hear the noise of a grenade bouncing into your room, you are quite confident that there's about to be a blam, and you take cover. If you were with someone who was not used to the horrors of war, and whose ears were not tuned to that particular clunking noise, they may be completely unaware that the grenade had entered the room. Thus, without warning, you drag them behind cover, then the grenade goes off. To them, it seems like you "saw" the bang happening before it happened. To you, it was just responding to something you had heard to avoid having to respond to the bigger blam.
Of course, this process can be tricky in the presence of unknown agents who may test the capabilities of this creature. They may actively seek out the weaknesses in this high sensory ability. They may not understand that you can hear the clink of a grenade hitting the floor, but they may figure out that wrapping the grenade in a few layers of cloth deadens the sound so that you can't hear it. Now you're on the same footing as everyone else.
To deal with this, the individual would need to reach out and change the environment in a way which makes it harder to surprise it. It may send out messenger particles to observe the world from further away and rely the information to you. Or, more primitively, it might choose a highly forested environment where it's difficult to maneuver without the organism seeing your trail (and potentially manipulating its own trail to try to lead you off).
We see this in human cultures. Consider the pattern of a series of waystations where one can exchange tired horses for fresh ones on a ride. This was done in countless cultures (if I remember, the Mongols and Romans both did this, and even in America our Pony Express was built on this model). By doing this, information of key activities, like troop movements, can be relayed to the capitol long before the troops could actually get that close.
Our nervous system also has several similar systems to give the illusion of prescience. Our patellar reflex is a reflex which tenses up our legs when we land from a height by sensing the tension in the patellar ligament (this is what the doctor bumps on your knee to check your reflexes). Upon receiving a shock, this reflex arc will start contracting the quadriceps before the nerve impulses from your feet reach your brain to announce that you have landed.
Half of this reflex is a hardwired monosynaptic link, carefully tuned. The other half of the reflex is under the control of the brain. The brain balances these two halves for different circumstances. The effect is a beautiful handoff from reflex arc to brain-innervated motor activity while your legs are compressing. Thanks to this beautiful link, it appears that our brain is responding to the fall before it gets word that the fall occurred.
Actually sensing something before light hits its eyes would be near impossible. However seeing faster is more plausible than other senses since the speed of light is so much faster than sound and other sources of information about the surroundings
Appearing to be able to see faster would also be difficult since other organisms observing the organism in question would not be able to observe what is happening quickly enough to make that assumption. For example, a person watching another person catch a ball that they hadn't yet seen arriving would rather be seeing the two things happening simultainiously.
As such the organism would to most other creatures to be impossibly quick and agile, but not much more would be apparent.
- One way of having an organism that sees faster is to have it process the information from the eye quicker. Maybe its sight bypasses what goes for the thalamus in it's brain kinda like the sense of smell do for us.
This would make it possible for the organism to percieve something quicker than other organisms who process sight differently.
The simple answer is that it can't be done, but the more complex answer is... well, more complex.
Jump to the next paragraph for more practical based stuff...
There is some dubious research out there which implies that humans begin to react to some things before they could possibly know of the thing they are reacting to. I didn't follow up on this at the time and I take it as very dubious at best. The idea, if I remember right, on how to explain this is something similar to how photons pass through 1 of the 2 slits in the double slit experiment, but doesn't know which one till it has hit, but if you interupted it at the point of going through it would be in one or the other which means that it has to have gotten the information from the aborted future or something like that. I don't buy it, but it's a interesting idea for explaining certain psychic phenomena.
The more realistic approach is that while it is impossible to see something before there is something to see (as far as we know) it is not impossible to be really good at predicting things based on all sorts of sensory inputs. This is partially how martial artists are faster than the normal person. It's not that they are truly faster, but rather they train theselves (indirectly) to react to things that they are seeing that tell what is going to happen next. You see a tightening of a muscle and you know, from experience, that it means a punch from x direction is coming and so you start to react to it, before it has actually happened.
This is also how and why we can sense things aroud us that we can't see. It's because we are noticing and feeling things around us that create a picture of our environment and whats going on even though we aren't able to see them.
Now combine the two along with the basic idea of the greater your knowledge of a situation the more easily you can predict things so that if you have 100% knowledge of all things in the universe you could predict the entire history of the universe from that point forward. What you have then is a way to "see" before there is something to see. Some martial artists display complete mastery of their environment, based mostly on trained instinct, if you were to give someone that trained instinct and with some set of knowledge I don't see why you couldn't produce the illusion of "seeing things before there is something to see"
The trick is to have senses that aren't strictly light/sound based. We already have such a device for sound waves: a laser microphone can "read" the sound waves striking a surface. Even needing to make the round trip, light is so much faster than sound that a laser microphone can "sense" sounds for a person long before those sounds actually get to that person (and, often, pick up sounds that would never get to that person).
Light, of course, is trickier, given that it's considered the universal speed limit. So, what we have to look at is, what's faster than light? Nothing!
So, how do we cheat?
Wormholes can collapse the distance between two points. If the distance from the viewer's eye to the target was "collapsed" in this manner, then the viewer could experience visual input before anyone else nearby. Depending on frequency and location of use, opening wormholes can have a number of... side effects.
Quantum Entanglement (also known as "spooky action at a distance") has been a popular haven in science fiction for this kind of effect (e.g., many "ansible" technologies). This spooky action has shown strange effects where "entangled" particles can influence each other instantaneously across distance. The problem with this is that entangling particles does not occur at a distance, the information transmitted is not at a macro level in the same manner as other sensory information, and the mechanics and limits of entanglement are not well known so any such solution will likely be subject to "science marches on" in a few years.
Alternative: time travel. If you were able to transmit neural signals backward in time to yourself, you'd be able to pick up "normal" sensory input, feed it to yourself X [units of time] ago, and it would appear that you received that sensory information long before you could have received it.
Technically, it is possible for things to travel at speeds greater than the speed of light in a medium (e.g. air, water) but lower than the speed of light in vacuum. c, the speed of light in vacuum, is the universe's speed limit, but like everything, light doesn't travel at the same speed everywhere.
See Cherenkov radiation, particularly visible in nuclear pools. Some particles goes faster than the light in water, which gives that distinct blue tint.
To make that work, you'll need three things:
A particle that goes faster than light in your medium (most likely air)
Creatures that emit that particle at that speed, which is definitely non-trivial, but could be explaining by some fashion of sorcery (e.g. every creature in the universe emits FTL radiation just because)
And of course, a creature that can sense those radiations/particles.
Another idea would be tachyons, which is a very hypothetical particle that always travels faster than light in vacuum. Tachyons are most likely non-existant in our world, because they break causality, so your world would have to operate on slightly different physics than ours to allow such a particle (and to allow it doesn't break the universe).