I find it very difficult to visualize spacetime curvature in my mind. When I look at the depiction like the one below it doesn't make much sense to me, instead, it is making it more confusing for me to understand spacetime curvature. In the depiction, it looks like that if someone would be standing at the bottom of the sun they would fall down into the spacetime curvature. But this does not happen. How should I visualize spacetime curvature? And is the reason that this is hard to imagine because spacetime is four-dimensional?
You are asking why it is so hard to visualize spacetime curvature, because it is four dimensional. In reality, it is very important to understand that the reason it is so hard to visualize is because our spacetime is intrinsically curved, there is no higher spatial dimension to move to, where we could look at the lower dimensions, and see the curvature extent into the higher spatial dimension.
Now intrinsic and extrinsic curvature are different. Extrinsic curvature is what is visualizable, and is on your picture, that bending grid. It is extrinsic because you are able to move to a higher spatial dimension, in your case the third, where the curvature on your picture extends to. In your picture, the grid is 2D, and the curvature extends into the third spatial dimension.
Intrinsic curvature is hard (of not impossible to) visualize in 3D, but we do have our imagination, and that is what we need to try to see how intrinsic curvature works.
Imagine the same sheet, but now you live on it, as a flatlander. Instead of the grids being curved into the third spatial dimension, lets bend them on the grid itself. The curvature now is inside the grid, and if you are a flatlander, and live there, you cannot tell they are curved. When you move along these lines, you still think you move straight.
Our spacetime is intrinsically curved, because we cannot move to a higher spatial dimension to see this curvature (the curvature does not extend into a higher spatial dimension, instead, we could say, it extends or creates effects into the temporal dimension), when we move along a geodesic, you are moving in curved space, but you from inside see this as moving in a straight line. This is embedded into our spacetime. The only way for us to know there is curvature is experiments for GR time dilation and gravitational lensing.
This type of curvature is what happens in general relativity. It's intrinsic not extrinsic. So to back to your question, you can't move behind the universe because there is no behind to move into. There are only the three spatial and one time dimensions - it's just that they are intrinsically curved.
The simple answer is: you can't.
You are an observer that only has freedom to move in 3 spatial dimensions and so it's impossible for you to actually visualise a fourth dimension and so trying to visualise spacetime curvature on the four-manifold is impossible. The 3D thought experiment is slightly better than the 2D however, and it might give a bit better intuition so I'll explain how I think about it below.
Imagine a 3D grid, with lines going left-right, up-down and towards and away from yo; kind of like lots of empty boxes stacked against each other to make an even bigger box. Now place the sun in the middle of this box. The lines of the grid will begin to curve under the influence of the sun and the lines in all directions will appear to converge into the centre of the sun. These lines are geodesics! So, if you don't apply a force then you will fall into the centre of the sun as you follow a geodesic. I know this may seem odd on the first read through, but it is the way I like to think about spacetime curvature.
However, if you really want to understand curvature on a fundamental level I'd highly recommend looking at some of the fundamentals of Riemannian Geometry and getting an intuition via that route. I can tell you, once I did my Manifolds course at Uni, the idea of curvature and spacetime really became a lot more clear. You'll get a sort of mathematical intution of curvature which is far more powerful than any visualisation I can try to draw for you.
But this is a good question, but unfortnately the answer is limited by our everyday experience.
The same way you see the curvature of the Earth on a flat map, through local scaling distortions of the map. Here is a map of a universe with positive curvature. The central galaxy is undistorted, but greater distortion is seen further from the centre. The "outermost" galaxy goes right round the map. I have explained this in my books. The diagrams are from Structures of the Sky
You can “undo” the scaling distortions on this map by mapping it onto a sphere, showing that the map would be the same whichever galaxy you choose for the centre.
Note that the sphere has no physical meaning. It is just a way of drawing a map. We can also draw maps of expanding spacetime, like this. The galaxies do not get bigger, but the distances between them gets bigger.
Other maps can be used. This one is exactly equivalent, but instead of the universe appearing to expand, galaxies appear to get smaller.