There: Refers to a place. Think "here." It is part of the word.
I placed my book over there or was it over here.
They're: Contraction "They are." Test sentence by replacing it with that phrase.
They're interrupting my reading, or They are interrupting my reading.
Their: Refers to possession. Think "heir."
It is their mistake, not mine.
When proofreading, do a find or search for each of these words. Then test the sentence. Never get caught by a troll again.