In my Chemistry class, we had a quiz and I helped another student cheat by showing him the answers on my paper but the teacher caught us. I know what I did is wrong and I know I deserve punishment, but I really respect my teacher and I don't want her to think less of me for this. Obviously that's already happened and there's not much I can do, but does anyone have any advice about how I can approach her and apologize for my actions in a way that doesn't seem like I'm trying to make excuses?
To your credit, I think you've already arrived at a good way to respond:
"I know what I did is wrong and I know I deserve punishment. I really respect you as a teacher and I don't want you to think less of me for this. Obviously that's already happened. I just want to apologize and not make any excuses."
Many schools have a centralized system for reporting academic misconduct, so that a student who accrues infractions in multiple courses will be held accountable. Just be prepared to accept that fact, along with whatever penalty is specified in the class.
Apart from that, if you remain enrolled in the course, be a good student. Answer questions, turn in your work on time, and be punctual.