You have to whittle down the boss's health bar until it's very low, like within an inch or less. Then when you throw the trophy stand at it -voila- instant trophy! I have Petey, Rayquaza, Duon, Golleum, and Tabuu so far. For some reason I can never get a trophy stand to drop down while I am fighting Pokey.
A tip on getting trophy stands to drop down more often is getting the sticker called "The President". It ups the chance of trophy stands appearing by +7. Someone had a good suggestion in another thread of putting it on whatever character you like to use, and then selecting them last in the character select screen. That way you beat down the boss to within an inch of its life with the other characters, then either get killed or kill yourself and switch to your trophy getting character and wait for a stand to drop. And sometimes you just get lucky anyways and a trophy stand will drop down before that.