Here's the best visual representation I was able to find, though I'm sure it's not perfect: http://www.relativitet.se/spacetime1.html
I'll try to sum it up as I understand it (which might be wrong):
1) The presence of mass distorts spacetime
(it's very important that it's spacetime, not just space). Why? I don't think we really know, we can just describe the result.
2) Objects naturally constantly move in a straight line through spacetime (the red and green lines in the pictures are
straight lines in spacetime's curved geometry, they just don't look like that way to us from our normal flat geometry perspective). However, since spacetime is curved, the projection of that straight line on our 3d space
is not a straight line, but a line that appears to be affected by a "force" of gravity.
So when the Earth goes round the Sun, it's just following a straight line through spacetime - but due to the way Sun's mass changes the geometry of spacetime, the projection of that line on just our 3D space is a curved orbit.