'it all went out the window' is an expression that means 'it disappeared/evaporated/dissipated.'
"They coped with everything until depression affected them, and then their coping mechanism "evaporated/dissipated/disappeared" (their coping mechanism went out the window).
Adding 'boom' to the front of 'going out the window is used to emphasize/stress the expression. It didn't *just* 'go out the window'/disappear, it very much went out the window.
'until their depression made an appearance' means until they were hit/affected by depression. Using 'made an appearance' is being symbolic or figurative. The speaker is saying '....until their depression 'showed up/arrived' or came upon them.