check out of (something)

1. To confirm that one has vacated one's hotel room (and pay the bill). We have to check out of the hotel in an hour, so you guys seriously need to start packing! The man's body was discovered when he didn't check out of his hotel room at the appointed time. Wait, we seriously have to check out of our hotel room at 11:00 AM? What are we supposed to do until our flight?

2. By extension, to complete the necessary steps that will allow one to leave a particular place. I can't check out of the office right now, but I'll be there as soon as I can. Did you scan your ID card? That'll unlock the door and allow you to check out of the records room. Make sure you set the alarm before you check out of the office for the day.

3. To complete the necessary steps so that one can borrow something from a system, such as a library. In this usage, a noun or pronoun is commonly used between "check" and "out." What books did you check out of the library this week? You'll need to check the equipment out before you can take it from the lab. I'm sorry, it looks like someone has already checked that book out of our branch. But I can see if any of the other local libraries have it.