when they say "flat" what exactly do they mean? are they saying that the universe has no thickness and that our perception of a 3 dimensional universe is only a perception or are they saying that the universe has some thickness but is relatively flat?

also, is the "fabric of space and time" the same thing as the universe but just at like... a different "level"?

A flat universe means that the spatial curvature is zero, (or close to zero) everywhere. Its hard to describe what this means in three dimensions, but in two dimensions, a space with zero curvature would be like a flat piece of paper, whereas a space with positive curvature would be rounded like a sphere, and with negative curvature it would look sort of like a saddle.

So in three dimensions, a space with zero curvature would be like R^3, a 3-dimensional analogue of the infinite plane. Positive and negative curvature is much harder to visualize in three dimensions, since the space would have to curve somehow into a higher dimension that we can't visualize.(Actually, curving into a higher dimension is not the mathematically or physically correct way of viewing this, for which manifolds with metrics assigned to them is used, but that's not important right now.) Flat universe has nothing to do with the thickness of the universe.

Fabric of space and time is usually used to describe space-time, which is the universe, and particularly when your are talking about the shape of space-time.