What does the old phrase "What happens twice, happens thrice" mean?
To my knowledge, that comes from a quote by Paulo Coelho. The full proverb is:
"Everything that happens once can never happen twice. But everything that happens twice will surely happen a third time."
Something that happens once is always something that happens once, by literal interpretation and definition; if it happens more than once, it isn't something that happens once.
Something that happens twice is referring to something that is not a one-time occurrence and therefore is not a chance occurrence. Because it's not a one-time occurrence, the proverb implies it to be a recurring one.
This proverb can be applied to endless situations and interpretations; it all depends on the mind that reads it. One example I've seen used more than others:
Girl and Boy are dating. Girl cheats on Boy with Man. Boy finds out and breaks up with Girl. Man starts dating Girl because she has only ever cheated once; there is no reason to think she will again. Girl then cheats on Man with Boy. Man finds out and breaks up with Girl. Boy will not date Girl because now she has cheated twice and most likely will cheat again.