The idea is that matter and energy curve space, but as we can't see space, it can be a unclear what is supposed to be happening. One way to see what is happening is to consider where something in space will end up when it's moving or standing still. The way this is often done in relativity is to consider the "light cone"of the object. I'm going to simplify this and talk about a "light angle". This is a light angle:
There is a single dimension of space, and time increases upwards. An object that starts at the centre of the space line and that isn't moving will stay where it is as time moves on. There are two diagonal lines in this figure. They represent two light rays which are sent off left and right. As time passes they will move further and further away from the central object at the speed of light. By convention the scale of these drawing is such that movement at the speed of light gives 45 degree angles between space and time, but the angle is not that important - what is important is to realise that points on the left and right lines at a given time are always the same distance as each other from the centre of the figure, the position of the object that isn't moving. Nothing can move any direction closer to the horizontal than the light ray lines as that would indicate movement faster than light.
What happens when space is curved by gravity is that this whole thing is twisted, like this:
This second figure is what gravity does to space and time if there was a massive object off to the right. By twisting space and time gravity leads to things that would otherwise be standing still moving towards the source of gravity. It's worth nothing that no force is involved! Someone falling under (gentle) gravity feels no difference than if they were stationary in space. Gravity is not a force, it's more like an instruction to things in space and time to move.