Hi, "Check out" often means that someone is looking at another person with interest. It could mean they are interested in a date (which is implied in your example) or it could mean that they are looking at someone for a specific reason (like "Check out her shoes!"). I hope this helps. :)

It can also mean to vacate and pay for one's quarters at a hotel. For instance, "We checked out of the hotel at noon."

If somebody ask me where did I borrow those books , can I say "I checked out those books from the library ." or "I checked it out from the library." (books are on my hands)

It really depends. As far as I know, check out could mean "look at"

Another meaning is to check out or to borrow items, like books from the library.