Poe is smart mouthed and watching him frustrate Kylo turns Hux on. He waits until Kylo leaves, and gives Poe something useful to do with his mouth.

Poe does not want this, struggles, mouths off, but Hux has all the time and resources in the world, and he always gets what he wants.

+1 if Finn is the stormtrooper on duty outside, figures out what's going on, and it's what makes him decide to get Poe out of there afterwards.