If the targeted creature’s owner has one or more cards in their library, that creature is put into that library directly under the top card.

If the targeted creature is an illegal target by the time Oust resolves, the spell doesn’t resolve. No one gains life.

Note that the creature’s controller, not its owner, is the one who gains life.

If the targeted creature’s owner has no cards left in their library, that creature is put into that library as the only card there.

If the targeted creature is a token, it will cease to exist after it’s put into its owner’s library.