Both forms are correct. They’re and their are an example of homophones, the words with different spellings, but the same pronunciation. Both forms are correct, but they serve various purposes in sentences. They’re is a contraction of a phrase they are, a present plural form of a verb to be. An apostrophe here suggests that it is a combination of two words. On the other hand, their is a determiner of a personal pronoun they, which expresses the possession or belonging of something. As we can see, these two have completely different functions are should not be mistaken.

Correct spelling, explanation: it’s a contraction of they are and expresses a state of being.

Correct spelling, explanation: it’s a possessive form of the pronoun they and determines a sense of belonging to them.

Definition of they’re:
verb: shortened you are; to exist, take place; occur;
They’re nothing more than friends, at least that’s what they say.
They’re living with his parents for now, but they want to buy their own apartment.

Definition of their:
pronoun: belonging or connected with them; also used instead of his/her;
Judy asked me to keep some supplies for a week, could you help me move their stuff to the garage?
It is their house now, they can decorate it however they want.