Page authority is essentially an estimated score that predicts how well a website will rank. The thought is getting a link from a website with a high page authority (trust) will help improve your rankings. Google looks at high page authority sites and trusts them more than lower PA sites. When one of those sites provides a link to you, they are saying you have valuable content and they trust you. When Google recognizes the link pointing to you page they trust you a little more, hopefully improving your rankings. It's like introducing your friend to another one of your friends. They don't know that second friend, but because they know you, they feel a little more comfortable around them.
Now with that said, page authority is only one component of SEO and I've been able to outrank higher page authority websites with lower page authority websites, so while it's nice to have, it doesn't necessarily impact rankings a lot.