What does ‘The vast majority (some 63%) of employees in the world are not engaged, “checked out”’ mean?
Solution 1:
Try googling "mentally checked out". As someone mentioned, it's related to checking out of a place like a hotel. Basically the way I think of it is, your mind has checked out of your head. "No one's home", in other (idiomatic) words.
Solution 2:
To "check out" is slang for a number of things, but in this case, it means to emotionally or intellectually detach oneself from something.
A cloakroom is a room in which hats, coats, luggage, etc., may be temporarily deposited (checked); where coats may be checked.
Similarly, as if you were checking your coat, here it means "checking" your engagement with your job somewhere else ("out").
All to say, that percentage is disengaged from their job. This would be in contrast to (and in between emotionally) the % who are engaged and the % who hate their work.
Solution 3:
To check out in the context you provided means the employees are going through the motions without much thought, and certainly not with passion or "fire in their bellies."
"How's Bill coming in his job, Alice?"
"Oh, I'd say he checked out long ago, Jack. Frankly, I'm surprised he's still in the same job."
"That's too bad, Jack. He had such great hopes when he was first hired."