There's a nice presentation of the paper here https://www.youtube.com/watch?v=-QjgvbvFoQA

In essence the effect comes from "precession" - the tendency of the flip to not be purely vertical but to have some wobble/angular momentum which causes it to flip in such a way as to spend longer on one side than the other. Depending on the technique this will have a greater or lesser effect on the fairness of the coin toss, ranging from about p_same = 0.508 for the best technique to one person in the study actually exhibiting 0.6 over a large sample which is staggeringly unlikely if the toss was purely fair. In the extreme, it shows in the video a magician doing a trick toss using precession that looks as if it's flipping but does not in fact change sides at all, purely rotating in the plane of the coin and wobbling a bit.

The video is quite a nice one for setting out how hypothesis testing works.