1. Void vs Vacuum
The first thing that needs to be done is to distinguish between void and space (ie vacuum).
Space is not nothing, because you can move things in it; think of it as the medium in which particles can move.
For if space was exactly nothing; then where could you put a particle? There is no place you can put it.
2. Crumpling space
The second thing is to imagine how you can warp this space; this is difficult using the space we actually live in.
So let's imagine a page torn out of Ryders QFT is our space. This is easy enough to warp - you can roll it into a cylinder, or crumple it in some other way.
3. Geodesics or straight lines
But how to do physics on this surface? Well, let's just take Newton's first law: a particle without any forces acting on it moves in a straight line.
When the page is rolled out flat on the surface of a table, the path this particle takes is easy enough to imagine - it's just the straight line we can draw by eye.
But how about when it's crumpled? Well, to make things easy for us let's imagine that the page has been crumpled and glued into a sphere. So how can we draw a straight line on this sphere? We can't do the obvious thing and just 'drill' through the sphere in the obvious straight line - because the interior of the sphere is not space but void - so a particle can't go there.
And we can't do the second most obvious thing either, which is just draw the straight line by eye on the surface of this sphere, because between any two points it's not clear what is the straight line that connects them - because to our eye they all look curved.
We turn this around, by asking if there is some unique property that characterises the straight line on the surface; well: on the surface of the paper spread out flat on the surface of a table, our original set-up, we see straight-away that the straight line between two points is the shortest path between two points.
4. Newton's first law again
So, we use this property on the sphere; and now it's easy to see how a particle will follow Newton's first law on this 'curved' or 'warped' space; it's still moves in a straight line, but here we call them geodesics.
In fact, this sphere is 'warped' in a way that the crumpled page is not; because if you draw triangles on it - which we can now, since we know what straight lines look like - we discover that their interior angles always add up to more than 180 degrees.
5. And GR, briefly
And this is how GR works: given a mass distribution in a space, this tells what the geodesics are in this space - ie how it's curved; and then particles move along these geodesics.