Jillian's Guide to Black Holes: Forming - Types - Outside - Inside - Finding - References - Websites
Sounds like it involves water, doesn't it? In a way tides are just a side effect of the namesake idea. This all has to do with how matter behaves in a gravitational field --- and I don't just mean by falling down! It's easier to explain this with a thought experiment.
Imagine there are two rocks high above the Earth, lined up vertically as shown on the left. They're falling towards the planet. The bottom rock is closer to the Earth which means it experiences greater acceleration than the top rock, which is further from the Earth. Consequently, the bottom rock falls faster than the top one, and they start to separate. This illustrates how tidal forces "stretch things out" in a radial direction from the massive object (in this case, vertically).
Now, imagine two rocks high above the Earth, lined up horizontally as shown on the right. The two rocks are attracted to the center of the Earth, so they accelerate toward it. As they fall, the two rocks move in straight lines directly toward the Earth's center. This makes the rocks move toward each other, since the center of the Earth is between them. This illustrates how tidal forces "squish things" in a tangential direction from the massive object (in this case, horizontally).
Wait, I'm standing on the Earth and I'm not being stretched out like those rocks! Thats true, but it's not a contradiction. The difference in distance between your head and your feet is very, very tiny compared to the distance between you and the Earth's center, so it's accurate to say that your head and feet don't really experience different accelerations (certainly nothing near like what those rocks were experiencing). You are considered "local", small enough that the acceleration due to Earth's gravity is considered a constant.
What's the catch? When you fall into a black hole, the curvature of spacetime is increasing, so the size of "local" rapidly decreases. When "local" no longer covers your whole body, you start to deform just like those rocks. Along the radial direction your body gets stretched apart (like the vertical rocks), and along the tangential direction (like the horizontal rocks) your body gets squished together. This eventually becomes so bad that your body can't take the stress and ... erm ... .well ... it painfully breaks apart. "Local" keeps shrinking, smaller and smaller, until it passes atomic size (ripping your atoms apart) and goes to zero, as the mathematicians would say.