The Einstein for Everyone website was a great eye opener for me.
Read the lectures in the "Non-euclidean geometry" and "General relativity" sections. It explains all without demotivating you with the hard math.
The key idea is don't try to imagine the curved space-time wrapped in higher dimensions it won't work. Just think in terms of converging and diverging "parallel" lines.
On Earth's surface (a sphere) moving along initially parallel lines eventually intersect (positive curvature). On a saddle like surface the initially parallel lines diverge (negative curvature).
Now let's put gravitation into the picture. Drop two bodies, that have vertical separation between them. As they fall the vertical distance between them increases. If you plot the action on a space-time diagram you see their world lines diverge, this means on the vertical direction the curvature is negative.
Now drop two bodies that have horizontal separation between them. They fall towards the Earth's center, so their separation will decrease, if you plot the action on a space-time diagram you will see their world-lines converge, so in horizontal directions there is positive curvature. If you sum up the curvatures you get 0, because there is no matter density outside the earth.
Now if you would drill hole through the Earth, and drop balls with vertical separations between them you will see that their wordlines will converge (as gravation is weaker inside Earth), so the curvature is positive, in all the 3 directions (since it doesn't matter where do you drill the holes). So if you sum it up you get a positive number, because matter density inside the Earth is positive.
What Einstein's equations describe is that the net space-time curvature at a point is proportional with the matter density at that point. It's easy to say but solving these equations are incredibly hard.