From a strictly grammatical point Lauren's answer is right -- you're talking about something that happened in the past, so "that was" is correct.
However, dialogue is often more colloquial and a first-person narrative can be more like dialogue than strict narrative. If you're trying to evoke the feel of the narrator actively telling a story to a room-full of listeners, you might find "that's when" to fit the style better. If, on the other hand, your narrator is dispassionately and remotely retelling past events, the way somebody might write an event chronology in a police report for example, he'd probably say "that was when". Whichever you do, be consistent -- if your narrator usually writes more formally, dropping a colloquialism in will stand out. So unless you want it to stand out for dramatic effect, I suggest avoiding mixing styles.