Anatoli-san, お世話になります is a phrase that means to become indebted to someone, but usually refers to the other person taking care of them. This is the kind of thing you would say to a host family, or even a friend who lets you spend a few nights at their place. After your stay, the phrase becomes past tense; お世話になりました. This can also be used for someone who has just done you a great deal of help, like if you were an exchange student, and someone takes you around town to get you hooked up with a bank account, cell phone, furniture and appliances for your apartment, etc.
Conversely, お世話にします/しました is when the speaker is saying "I will take/have taken care of you," in so many words. I don't hear this one very often, though.