Logic behind this?

Well, for one, your MMR helps determine how much LP you get per win. If you have high MMR compared to those in your division, you get more LP per win so that you will rise in rank faster.

Leagues (and Elo) are a display of your current position on the ladder. MMR is a measurement of your current expected skill level.

MMR dictates who you play against. Leagues are only used for seasonal rewards.

Was meaning why have two different things that is supposed to measure your skill level. Wouldn't be better to just use mmr or just use divisions?

It's so the people in Bronze 5 that suck even more than people who never played this game before (no seriously, go watch Kshway videos, or Bronze spectates) are separated from Bronze smurfs that do climbs out of Bronze for fun and people who are doing their best trying to get out