Here's the way I understand it:
The Archmage's Staff sits out and has a slot that can be placed on like a normal room. The effect of this slot, at resolution, is "take control of the archmage's staff" (move it in front of you).
The next round, while it's in front of you, you can use its effect as an action, just like if it was a vault card (or a 0-cost spell). Other players (or yourself, though there'd be limited reason to) can place in the staff's slot, and if someone else has a mage there during resolution, they'll take the staff away from you.
this is also the way I understand it. although there is a reason to place your own mage their if you own the staff and that is to make it more difficult for other players to take.