"Check out" has 7 definitions - 5 are formal, and 2 are considered informal.
1. When you leave a hotel or a hospital, usually after paying the bill.
"Are you checking out?"
"We have to check out at 11am tomorrow morning."
2. To examine something to learn more about it.
"Check out my new website!"
3. To examine information to be correct, or true, or satisfactory.
"The police is checking out the witness' story."
4. To examine something in order to pass a test.
"This machine checks out, so let's move onto the next one."
5. To get examined at the doctor's.
"If you're not feeling well, you should get yourself checked out."
6. (informal) To assess someone's physical appearance.
"Check out that guy over there."
7. (informal) To be mentally or emotionally absent, either from tiredness, frustration, hopelessness or a loss of interest.
"On the last day of the project, she completely checked out because of all the stress."