Your MMR will stay the same. It is not known what happens to MMR confidence, but it may well just not come into play, as the game will need to place you somewhere (a low confidence value does prevent you from being promoted, even when your MMR is already in another league).
So, in general the purpose is to force re-placement into a league, if you believe you're not where you belong. That is when you're tired of waiting for a promotion, or, arguably more importantly, when you are too high. Since some recent version it is impossible to be demoted mid-season, only to re-qualify to a lower league.
On one dull Friday night, after several vs.AI matches and some three liters of beer, I had a brilliant idea - go qualify in 1v1, so I will get placed low and play slow and peacefully. I was pretty drunk, and played Random. After four proxy 2-gates and one 8-pool vs a Master leaguer, I was left with Hot Shot, in Platinum, and an expression like O_O.
After getting brutalized in Platinum for a couple of days, I've left a league and found my place in Silver, and all was well once again :-)