Could someone please explain how thermal energy ties into work equations and energy equations? My teacher gave us some homework with this in it and then only briefly explained how to get it in those specific examples, but I still do not get it.

You've asked a very broad question, and it would help to know specifically what concept you're struggling with. Can you post a sample question or problem that you need help on? And when you do, please tell us what you've attempted and which part of the question is troubling you (don't just post a homework and ask us to solve it for you). Then we can help you out.