Why Balmer series exist?
The entire process has two parts: Excitation and relaxation. The first is clear enough, we excite an electron into a higher energy level.

Now this electron wants to fall back down. There is no roadsign or intrinsic selection rule that requires the electron to immediately go back to the ground state. Any state in a lower shell will be less energetic and by chance electron 1 happens to stop in state A while electron 2 of a different atoms stops in state B. Any electron that is not in a ground state will remain there — probably for a duration of about a few femtoseconds or less — and then relax further. The universe is not deterministic so nothing is telling the electron it has to take the fastest path.

Note that this gedankenexperiment does not make any prediction which path is most likely. You would need more theory (that DavePhD is likely to give) to explain that.