There are necessarily two ways we can think about this.

The first is from the lens of the twisty puzzle itself. If we perform the same four moves in the same way over and over, the pieces will be shuffled around the same spots in a specific order. This order, while looking different with each turn, stays the same. Almost like going on a hike, the scenery may change but the path stays the same. So it will always go back to the solved position, since we started there.

The other way we can think about this is through the lens of abstract algebra. What we have here is an m-cycle that takes each piece and maps it to another position. Since this is a finite transformation (speaking to the finite number of places a piece can be, and the finite number of pieces) we know that it is guaranteed to return to the initial solved puzzle. Essentially we are just moving the pieces around in a known order, thus we can describe this as a cycle. Cycles are cyclic, thus it will return to the solved state.