The wikipedia article is a bit difficult to understand. In a nutshell, you can think of a Rainbow Table as a large dictionary with pre-calculated hashes and the passwords from which they were calculated.
The difference between Rainbow Tables and other dictionaries is simply in the method how the entries are stored. The Rainbow table is optimized for hashes and passwords, and thus achieves great space optimization while still maintaining good look-up speed. But in essence, it’s just a dictionary.
When an attacker steals a long list of password hashes from you, he can quickly check if any of them are in the Rainbow Table. For those that are, the Rainbow Table will also contain what string they were hashed from.
Of course, there are just too many hashes to store them all in a Rainbow Table. So if a hash is not in the particular table, the hacker is out of luck. But if your users use simple english words and you have hashed them just once, there is a large possibility that a good Rainbow Table will contain the password.