A chain wear indicator tells you if the chain has stretched. You shove it in one link and see if it will fit in another. If it fits the chain has stretched and needs to be replaced; otherwise the chain is ok.
It doesn't do anything you couldn't do with a ruler, but it is idiot proof (no offense, I have one). After the tires and break pads (which wear out by design), and brake and shifting cables, the part of a bike that needs the most attention is the chain. Because of that and because a chain wear indicator is so easy to use, it should be in the intermediate tool kit, not the advanced tool kit.
If you ride with a stretched chain you will prematurely wear out your chainrings and cassette (which cost a lot more than a chain). If you let it really go, it can slip. This will of course happen when you are stomping on the pedals or hitting a bump or both.
Note that a chain may need replacement when there is as little as 0.75% stretch. Most chains have a least 100 links, so it is easy to see that, there is almost one new link increase in chain length for this amount of stretch. But your bike will ride just about like a new one with a new chain.