How does gravity truly work?

Space (better space-time) is not like a trampoline. That's just a really poor image that is mostly used in pop-science, even by people who know better like Kip S. Thorne. I did hear him use a much better analogy once in a talk for physicists in which he compared gravity to a river:

If you want to stand still while swimming in a river (relative to the banks of the river), you have to keep swimming upstream (use your rockets...) or be anchored to something (stand on the surface of a planet). As long as you are swimming with the stream, the water doesn't exert any forces on you, but if you want to stand still relative to the banks (or swim upstream), then there will be a significant force against the flow of the water.

The analogy for a black hole then becomes a circular waterfall: you can drift all the way to the edge of the falls and keep swimming there until you are tired (your fuel runs out...) and even swim back upstream (if you brought enough fuel...), but as soon as you fall over the edge, there is no swimming back! Over the edge of the falls the world at the level of the river disappears and it's only down from there... towards the level of "the singularity", that's the bottom of the falls that will crush you.

The model is not perfect, of course (it doesn't work for orbits), but it motivates that spacetime (which is the real foundation of gravity) has a one way component: time.

Now... in reality all of this is a lot more complicated and you will have to learn general relativity, if you really want to understand the details. The good news is that the golden time of gravity research has just begun. If you decide to become a physicist, you will see absolutely amazing things in your lifetime and many may be related to research on space, time and gravity.

I still don't understand gravity because if space were like a trampoline, then earth would be spiralling in towards the sun along with all the other planets, right?

The trampoline analogy is in two dimensions. A small comet does spiral and end up in the sun, because as it moves it is losing mass and therefore angular momentum. Angular momentum is what keeps the earth and planets from spiralling down to the sun, in stable orbits, because angular momentum is a conserved quantity. To decrease, the earth should be losing mass, which does not happen in the vacuum where the earth moves. On the trampoline, because of friction the little balls lose momentum and angular momentum and end up spiralling to the center, no stable orbits can be established.

People developed the theories of how gravity works over centuries. Observations showed that what goes up, must come down, and that created the concept that gravity is a force, similar to the force needed to pull or push somebody. Newton made the first mathematical formulation which explained not only falling apples and throwing javelins but how the planets revolve around the sun, by assuming a force felt between the mass of the sun and the mass of the planet. This works very well as a mathematical model of gravity. The trampoline goes as step further, trying to model the gravity as expressed in General Relativity; making the masses generate the space and time , everything then follows mathematically.

To really understand physical reality with the available models, one has to study hard, mathematics and physics, which seems a big task for your age group, although I know of people who started on the physics path at 14. Mathematics is important in order to be able to understand physics .