nobody notices your mistakes as much as you do
I think the intended meaning of this quote is simple -- don't be afraid to make mistakes, and don't stress out over them when they do happen. However, I took a different and more trombone-related meaning from it.
When you're practicing, no one can hear your mistakes except you*. So no one can fix those mistakes except you. There's no one to point them out, no one to suggest corrections, no one to stop you mid-phrase and go HEY WHY DID YOU PLAY IT THAT WAY?!
You're it. So maybe the message is not to stress out about mistakes, but another message is not to ignore them either. I'm very guilty of this. I'll flub an articulation or play a rhythm wrong, and think to myself "Oh, I know how it's supposed to be" and keep going. And not fix it. What does this accomplish except giving me reinforcement to make mistakes?
So -- yes, nobody notices your mistakes as much as you do. That also means that it's up to you to fix them.