edit - honestly not a troll. is it the specific formatting of "em" dashes? i know for sure we use them all the time. or at least i do. but they're just dashes to me, so..
Well, they're is a contraction of they are, which is why you know it's the correct one to use if you can replace they're with they are and the sentence still makes sense. The word their is possessive so if you're talking about someone or even something possessing something else, you would use their. There is in reference to something or somewhere else.
I can't remember the specific rules I was taught in school, but I still know the correct usage many years later.
There was a snake over there, they're trying to find it now, cause it isn't native and none of our friends say it is their snake!