Why C? And why not the others?

Correct answer: C. They're going to meet us over there, and they're bringing their dog.

These three words sound identical but do three completely different jobs. 'They're' is a contraction of 'they are', so it fits both times this sentence describes an action the group is doing ('they are going' and 'they are bringing'). 'There' refers to a place, so it correctly describes the location being met at. 'Their' shows possession, so it correctly describes the dog belonging to the group. The correct sentence uses all three in exactly those roles. Every other sentence swaps at least one of them into the wrong job: using 'their' or 'there' where an action is meant (instead of the contraction 'they're'), using 'their' or 'they're' where a location is meant (instead of 'there'), or using 'there' or 'they're' where ownership is meant (instead of 'their'). Because all three words sound exactly alike when spoken, the only way to tell them apart is by checking which job — action, place, or ownership — the word is doing in the sentence.

Source: BBC Bitesize KS3 English: Commonly confused homophones — there, their and they're