Thaumaturgy basically takes ways to go from having a bunch of gems you don't want and going to ones you do.
I'm a particular fan of Cold and Shock effects on weapons. Suppose though, that while I can find Cold-ember relatively easily, I cannot seem to find any Flow-Ember. Thaumaturgy would allow me to make Flow-Ember from my supply of ember of other colors.
Here's how you read the picture: Put your finger over the gem you want. The resulting 4 gem types are what you need to put in to get it. They have to be all the same grade (All Cracked, for example) and will produce one gem of the next grade higher (to continue with the example, Dull gems)
To get a Dull Flow-Ember, Vanilla Torchlight requires 2 Cracked Flow-Ember. Thaumaturgy allows you to put in 1 Cracked Fire, Cold, Life, and Pure ember to get it. This is a functional change to the transmutation system.
Arithmancy, on the other hand, merely allows algebraic equivalents of the already present in Vanilla Torchlight's Transmutation formulae to work.