I found the creator of Rainbow Table's paper, aimed at cryptanalysts, was pretty inaccessible considering the simplicity and elegance of Rainbow Tables, so this is an overview of it for a layman.

Hash functions map plaintext to hashes so that you can't tell a plaintext from its hash.

If you want to find a given plaintext for a certain hash there are two simple methods:

• Hash each plaintext one by one, until you find the hash.
• Hash each plaintext one by one, but store each generated hash in a sorted table so that you can easily look the hash up later without generating the hashes again

Going one by one takes a very long time, and storing each hash takes an amount of memory which simply doesn't exist (for all but the smallest of plaintext sets). Rainbow tables are a compromise between pre-computation and low memory usage.