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The normal condition of life in a planet (Earth right now) is that below us is the planet's core, and above us is the sky.
Now if there's a hollow planet like above, is that possible? How does gravity work on a planet like that?
Since the planet is not fully closed shell, I don't really bother about light from its star, but there's a bonus if the answer include that.
In general, the problem with hollow-Earth setups is that objects in hydrostatic equilibrium cannot be hollow - and planets must be in hydrostatic equilibrium. Planets form through collisions of smaller pieces of rock and dust, and eventually accrete enough matter to become substantially large. There is absolutely no way for a planet to form with a hollow interior; it should have been compacted during its formation.
In a typical hollow world that is fully enclosed, there should be virtually no net gravitational force on a person inside. For a spherical mass distribution, the shell theorem states that anyone inside a spherically symmetric shell should feel no net gravitational force from the shell. There will, of course, be variations in mass density, but they wouldn't add up to anything substantial.
A semi-open shell means that the shell theorem is not fully valid, but it does seem to imply that there will be very little gravity. Your image shows the planet about two-thirds enclosed, meaning that most of the field will be canceled out. Certainly all vertical component will be, assuming the two shells are identical.
Additionally, having the shell only partially enclosed means that the atmosphere has a chance of escaping to the outside of the planet through diffusion, where it will indeed feel a net force - centered on the center of mass of the planet, which is probably the center of the shell. There would still be some left inside, I would assume, but much could escape to the outside. I don't know how significant this could be, though.
Jason K suggested using rotation to keep things on the inside surface of the planet inside, via the fictitious centrifugal force. Here, the magnitude of the centrifugal force is the same as that of the centripetal force at the equator - the place where it is strongest - is $$|\mathbf{F}|=F=m\omega^2R$$ where $m$ is the mass of an object, $\omega$ is the angular speed of the planet, and $R$ is the radius of the inside of the shell. The magnitude of the angular acceleration is then $$|\mathbf{a}|=a=\omega^2R$$ Assuming that the planet rotates at the same speed as Earth, and its inner surface has the same radius as the Earth, then $$|\mathbf{a}|=\left(7.29\times10^{-5}\text{ rad s}^{-1}\right)^2\left(6,370,000\text{ m}\right)=3.39\times10^{-2}\text{ m/s}^2$$ That's at the equator, where the force is at its greatest.
The planet would need to spin about 17 times as fast or have a radius 289 times as large in order to have $|\mathbf{a}|=g$. It could spin at about 10 times Earth's speed and generate Martian-level surface gravity, though, which would be decently suitable for humans. Apparently, the fastest spinning exoplanet known, $\beta$ Pictoris b, spins at roughly three times Earth's speed.1 That means that to generate Martian gravity, the planet would have to have a radius about 11 to 12 times that of Earth - making it the size of Jupiter! For what it's worth, measurements place $\beta Pictoris b at $\sim1.46$ Jupiter radii. No terrestrial planets will get anywhere near that large.
1 That said, we haven't measured the rotation periods of many exoplanets.
A hollow sphere structure is still possible, theoretically.
If the shell of the world is thick enough and strong enough, then it could withstand the stresses of rotation, and possibly gravitational stresses of surrounding celestial objects. This is also the same basic structure of a Dyson Sphere.
Gravity is produced by mass, it doesn't matter what shape it is in. Therefore, a solid sphere of sufficient mass will produce gravity from the center. A giant planet sized pyramid of mass would produce gravity from the center, which would be feel stronger on the planes (closer to the center) and weaker on or near the points. A giant cube of mass would have a similar effect. Things get a bit strange with a hollow object.
A hollow sphere will produce gravity not from the center of the sphere, but from the center of the existing mass, which in this case is material making up the shell of our hollow sphere. For the sake of theory, let's say our hollow shell-sphere is the size of Jupiter, with no mass at all in the center. The thickness of the sphere let's set at the same as the diameter of the earth. If gravity has the same effects, then what would happen is that the middle part of the shell would melt, and eventually compress to metal as the heavier elements precipitate or percolate out of the molten material.
Note that this would happen everywhere throughout the shell-sphere. The theoretical end result would be a shell which is comprised of an inner layer of rock, under which is a layer of mantle, under which is a layer of liquid rock, and finally a "core" of metal, only this would actually be a sheet of metal. On the other side of the sheet of metal would be another set of layers, liquid rock, mantle, and then outer surface of rock.
This would then produce 1g across the entirety of both inside and outside surfaces of the sphere (assuming, of course, that this structure was actually somehow strong enough to retain its shape). However a sphere like this is most likely to be a complete sphere, with only small openings, unlike your picture.
Gravity on the object in your picture would draw towards the greatest density of mass, in other words, the top and the bottom. This would mean you could stand on the outside or inside of the top or bottom half, as well as stand on the outside or inside of either sphere. Gravity would be less towards the edges, as the amount of mass is also less.
Unless there are external or additional forces not depicted, there is nothing stopping the two halves from exerting gravitational attraction and collapsing into each other. Not to mention the seemingly free-floating islands near the edges of the top and bottom.
However, as you did not use any of the Science tags, you could postulate magic, handwavium science, other unknown elements, forces, or environments which would in fact support and preserve such a structure as depicted.
Such a world would be placed under extreme gravitational stress if it were to exist in a solar system by the sun and other planets (unless again, you opt for magic or handwavium). Should it exist in a traditional solar system, then the light of the sun will only reach certain parts of the world, casting much of it into shadow most of the time. While that could be thematically and plot-wise an interesting aspect, it would certainly limit what kinds of life (flora and fauna) could successfully exist and adapt.
If the central pillar-like mountain structure was composed of or covered in a highly reflective material, that would alleviate some of the lighting issue, but would also make day/night cycles rather complicated. Also, note that if the orbital path was oriented so that the open middle part was in the plane of the elliptic - with no axial tilt, then there would not be night in many places of the world, only periods of direct light, and indirect light.
Reality Check
Before I start anything, that planet in the picture will not work, period. But... There is a way for a planet like that to be made.
Magnets. You know how magnets work, right? North end comes to south end, but north+north or south+south won't work. Well, that's how this planet will work. Extremely powerful magnets. You would have a mainly north (or south) magnet sphere in the center of your planet, while the inside shell (crust) would be north also (or south).
Gravity
Gravity, might be tough. In fact, gravity won't exist unless you have the planet spinning VERY fast, so stuff is pushed down to the "ground," or you could have your planet in the middle of tons of starts in a very rare (most likely impossible) orbit so that gravity from starts will push your animals towards the ground. Or, you can have that your creatures on the planet would have south magnets in there feet, and north magnets in their heads.
Now, here's the thing, this planet will not be able to
Have open air to space,
Form naturally.
Lets go on 1. the air and oxygen and all that stuff will be sucked out, maybe not sucked out like you see in space movies, but, if this planet just goes and appears. Then yes, like the movies.
and 2. This planet will not form naturally, because the chances are SMALL. its probs in the range of googleplexionths.
But, its your world, so make it anyway you want.
Gravity is based on mass. it does not matter if a planet is hollow not, it still has mass. The hollow sphere makes the math harder, the concept is the same. It doesn't matter one bit if the planet is enclosed or not. If the hollow sphere below you has the same mass as the Earth, you would experience gravity if normal force.
One wrinkle, however, is that you would now have a thin shell instead of a solid sphere, and this could create unusual gravitational gradients of the sort what would keep Larry Niven up at night...
Taking for granted that the inhabitants of this world have reached an extremely advanced stage of technological progress, knowing that the only limit to their technology is that they can't violate or circumvent lightspeed, so no space colonization, they could try a most massive work of planet engineering, using the planet itself as construction material to progressively hollow it out for them to make a working ecoystem inside it, taking warmth and heat from the core. Their world would make for the nucleus of a titanic nuclear reactor/sun.
The picture you provided should be shown sideways, so that, as the world keeps spinning, it provides artificial gravity. All other resources, water and food, would come from the surface -at this point, given that these people basically restructured their planet, covering the surface with greenhouses would be the least of their problems.
Titanic compressors could provide fresh air from the outside while inside the CO2 is worked by genetically modified vegetation.
Gravity graduates to zero both toward the mid point through the outer shell, and on the inside surface towards the center point. You could actually fly around in the middle, but you might wind up like Icarus.
As already hinted by other answer, it is almost impossible for a "natural" planet to result as you asked.
My idea: a planet-sized generational ship, carrying milions of people toward another galaxy (or a very far star in this galaxy).
After some kind of disaster (or for a specific choice of the builders of the ship), the people inside lost consciousness of living inside a starship, and formed a primitive society. Luckily the machinery that moves the ship is completely automated and self-repairing (or could be operated by a group of technicians who are not allowed to have contacts with the people living in the sphere).
People would then live in a giant spheric biosphere on the tip of the ship and since this ship is in perpetual acceleration, it would create a gravity going from the ceiling of the sphere toward the bottom (similar to your drawing).
Some drawbacks of this idea:
* a ship with a spheric biosphere on its tip is not very efficient as calpestable surface/volume ratio. Such a design would be better with a cylindric-shaped biosphere.
* if in perpetual acceleration, this ship would likely travel almost all the time at almost light-speed, which poses a temporal problem. Even if reaching another galaxy would require milions of Earth years, at relativistic speed the time inside the starship would be slowed to the magnitude order of some tens-hundreds years. If your world needs ancient myths (or enough time for people to forget they are in a starship), the travel could be too short
