Here's a nice explanation from /u/gameryamen on reddit:

Say you have a flat arrow pointing up. You spin it 3/4ths of a rotation clockwise, so it's pointing to the left. The simple way to undo that rotation (meaning, get back to the starting point) is to simple rotate it counter clockwise the same amount. But another way to do it is to rotate it 1/4 of a turn clockwise.

Another way to describe that last 1/4 turn is as two 1/8th turns, right? We're scaling the amount of rotation down, then doing it twice. The factor we need to scale down by is pretty easy to work out in this simple example, but it's much harder when you're working in 3D, and working with a sequence of rotations.

However, this paper shows that for almost all possible sets of rotations in 3D space, there is some factor by which you can scale all of those rotations, then repeat them twice, and you'll wind back up at the starting position. A key thing here is that we still have to find or calculate what that factor is, it's going to be a very specific number based on the set of rotations, not any kind of constant.

Why does that matter? Well, besides just being a neat thing, it might lead to improvements in systems that operate in 3D spaces. Doing the two 1/8th turns takes less work than doing a backwards 3/4ths turn. Even better, it allows us to keep rotating in the same direction and get back to the start. If calculating the right scaling factor is easy enough, this could save us a bunch of engineering work.

Edit: The most common question is "why do two 1/8th rotations instead of just one 1/4 rotation?" The reason is because the paper deals with a sequence of rotations in 3D, not a single rotation in 2D. But that's kinda hard to wrap your head around without visuals. This is going to be a little tortured, but stop thinking about rotations and imagine you're playing golf. You could get a hole in one, but that's really hard. A barely easier task would be aiming for a spot where you could get exactly halfway to the hole, because you could just repeat that shot to reach the hole. There's still only one place that first shot can land for that to work, it still takes a lot of precision.

But if you change your plan to "Take a first shot, then two equal but smaller shots", there's a lot more spots the first shot could land where that plan results in reaching the hole on your third shot. Having one more shot in your follow up acts as kind of a hinge, opening up more possibilities. This is what the "two rotations" is doing in the paper, it's the key insight that let the researchers find a pattern that always works.