Naruto can no longer pay the rent, as he has a very large debt for not paying previous months and is going to be evicted.
Tsunade owes Naruto money, but she lost it at the casino, so seeing herself out of options, Tsunade invites him to live with her until he can pay his rent again.
Tsunade, being a very disorganized person, sees an opportunity to exploit Naruto, as she uses him for everyday tasks, trying to squeeze every last drop out of him for her own benefit... and she's not the only one in the village who will do it.