Linear congruential generators (x = k1*x + k2, unsigned with truncation on overflow) have some strange properties, some of which the paper explores. If you repeatedly take three sequential values from one and treat them as 3D coordinates, the points line up in parallel planes. (There's an explanation of why in Knuth.) Something has to be done to destroy that order. The solution here is to XOR the upper and lower halves of the 128-bit state to get 64 bits, then circular shift that by the high 6 bits of the 128-bit input. This passes most of the classic tests for random number generators.
The paper is really two papers. The first paper is a thoughtful and careful explanation of how we should consider judging pseudorandomness, and how to measure various criteria, along with a survey of many popular ten-lines-or-less PRNGs which PCG competes with. I found it interesting.
That's the paper, basically.