When you leave something up to someone, you are giving that person the responsibility and you are trusting that person to do the job. That something can be any activity; it can be planning a party, writing a letter, talking to your parents on your behalf, finding a cheap plane ticket, anything. You are basically telling this person to take charge, and you believe that he or she will do a good job.
You can also tell a friend to leave something up to you, which means you're telling your friend not to worry and that you will take care of business.
A: I want to stay in New York longer, but I don't think my parents would agree.
B: Just tell them why you want to stay longer. I'm sure they'll understand.
A: I don't know about that. They were pretty adamant that I go home after this semester.
B: Why don't you leave it up to me. Sometimes, it helps coming from a friend. I'll make them understand.
A: Oh, thanks. I appreciate it. I think they'll listen to you. ... So what are you guys having for dinner?
B: Well, John's in the kitchen making something. It's his turn to cook, so we're leaving it up to him.
adamant - strongly insist
Alrighty folks, what sort of decisions or activities do you trust to others? Do you trust someone else to pay the rent or do maintenance on your home or talk to clients? You can say, "I leave that up to my roommate (or my landlord or my business partner)."
Don't forget to practice.