I think I can explain it:

Other side: 1/2 flip, 1 1/2 flips, 2 1/2 flips...

Same side: 1 flip, 2 flips, 3 flips...

It seems like there's equal chances, but my theory is that the 1/2 flip is the least likely thing to occur. When you take that into account, there's a slightly increased chance that it's going to land on the same side.