"고양이가 자러 집에 가요."
Translation:The cat goes home to sleep.
I realized, especially with this lesson, that removing the new verbs (with new conjugations) will make things clearer. Starting off the lesson, I always get confused which goes first.
For this example, 고양이가 자러 집에 가요. If I'm not sure, I will remove the verb with the new conjugation (자러), and I am left with 고양이가 집에 가요 - The cat goes home.
after this, you can put back the weird-ass conjugation (자러) and analyze what it means. ~(으)러 means "in order to", and 자 is sleep (from 자다/잠자다).
So keeping intact the original meaning of the sentence, The cat goes home + in order to sleep = The cat goes home to sleep.