When you play this card, must you use its power immediately?
Does that action count as a play?
Does the action happen every time it is your turn or just once and that’s it?

A: The text says “If you have this on the table, once per turn you may move another of your Keepers to another player, then discard one of that player’s Keepers.”

Keepers or other cards with powers that let you do extra stuff on your turn are optional Free Actions you can take at any time once during your turn. It is not required, and it does not count as a play. “Once per turn” means you may do it once, each time your turn comes around.

So yes, you could choose to take this action immediately on the same turn that you lay it on the table, but as the word “may” indicates, it’s not required. You can choose to use it each time your turn rolls around – or you could choose not to.

Think of this action as the Catapult flinging one of your Keepers (like, say, a Cow) onto someone else’s collection of Keepers, landing there, and squishing something (like, for example, a Trojan Rabbit).