Like many other things, milk contains bacteria, the tiny little critters that live naturally in and on lots of things and are so small you can only see them if you use a microscope. Milk also naturally contains a kind of sugar known as "lactose." The bacteria that live in milk get energy from this special sugar and use it to reproduce to make even more bacteria!
When the bacteria use the lactose sugars to reproduce, they change it from "lactose sugar" into "lactose acid," which tastes sour. When this happens, the milk curdles and is no longer the smooth, tasty liquid that you're used to drinking!
While some bacteria are good, there are some that are not so good, and the kind that live in old milk can be among the ones that are better to avoid. After all, you'd probably rather have cereal be the only thing in your milk that isn't smooth and creamy!
Further Information
The milk we bring home from the store has been pasteurized, meaning it's been heated to high temperatures to kill a lot of its bacteria. As long as milk is kept cold, it usually takes some time before the remaining bacteria in pasteurized milk will make it go sour. (Most milk is sold with an approximate date of how long you can keep it before this will happen.) The natural souring process of milk isn't always such a bad thing; in fact, it's how certain other dairy products like cheese and yogurt are made! However, unlike old milk, those products have usually been made using special bacteria that won't upset our stomachs or make us feel ill.