In general: I explain the outlines of the way it works on this page (http://www.lthr.xyz/programming/13/6/2015/eternity-ii-solver...). This is the technical aspect. But because not everyone likes the technical details, I wrote a parable on this very subject (Theseus, Penepos [a name I created to anthropomorphize P?=NP {PeNePos}], and the Eternal Labyrinth: http://www.lthr.xyz/blog/7/6/2015/theseus-penepos-and-the-et...). This is a core aspect of my project.
You will probably think that I am crazy, but my goal with my program is to simulate, as subtly as possible, the idea that Penepos symbolizes.
But wait, TetraVex puzzles are NP-complete (http://www.sciencedirect.com/science/article/pii/S0020019006...), and my algorithm solves large ones almost as fast it is possible to check them, so maybe Penepos is not just a fantasy... (You may understand with this sentence that I am a strong believer that P = NP).
I have updated my algorithm (when I worked on the step-by-step visualization process, I realized that sometimes it was doing worthless extra steps), then it is faster than the first version. The new version is online. You can also upload your own puzzles now.
For 16x16 TetraVex puzzles with 23 different colors, it takes, for about 80% of them, less than 10 milliseconds on a regular computer with no parallelization (I use two instances: one on a Surface Pro 3 and another on a EC2 t2.micro).
For the others, it takes about 100 to 200 milliseconds to solve.
Then, IMO, it takes 'almost' the same time to generate a solution than to check it.