In the real world, space curvature happens when you get something
really massive in one place. Actually, this occurs in any
gravitational field, but it's often too small to be of any consequence.
Did you ever play with one of those big yellow coin funnel things, the kind you usually find at a museum? You know, the one where you put a coin on a slide, and it goes down onto the funnel and spirals inward, faster and faster, until it's going so fast that it looks like it's defying gravity by not tipping over or falling down the hole? That, more or less, is a demonstration of how gravity bends space; specifically, you're seeing how a black hole works.
Let's go back to our sheet of paper, but let's make it out of rubber this time, the really stretchy kind. Then, put a bowling ball on it. It would sag down a lot, right? And probably stretch, too. Let's take the bowling ball off for a second, and draw a line about halfway between an edge and the center. Then put the bowling ball back on. When the rubber stretches, it looks like the line has been "tugged" towards where the bowling ball is now resting, doesn't it? Well, that's a geodesic for you. The light is traveling along a straight line, but it looks curved because gravity has warped the space. A black hole is just so massive that it stretches the rubber down infinitely far, until it tears a hole in it, but hey, that's another node entirely.