So how exactly does this ability work?

The process starts as you look at or reveal cards from the top of your Main Deck due to an effect. Let’s use Stacked Deck as an example. You play Stacked Deck and look at the top 3 cards of your deck. Nocturne is among them. You now get to choose whether to banish him or not. If you choose to banish him, you immediately begin the process of playing him from banishment, placing him on the chain as a pending item. Then you continue resolving Stacked Deck. There are only 2 cards left of the original 3; you’ll choose one to put into your hand and recycle the other one. Then, since Stacked Deck has finished resolving, you’ll finish the steps of putting Nocturne onto the chain from banishment, paying [A] (that’s one power of any type) instead of his base cost. As usual for units, he’ll resolve immediately and enter the board exhausted. If you weren’t able to play him for any reason, he’ll just stay banished.