New to chess, why does game analysis say that this was a bad way to defend the rook?

Your move was good, it was just not the absolute best move. The analysis shows that exf6 is best because if black now captures the rook Qxa1 you can play Bc3 at this point and it traps the queen. So effectively you did not need to defend the rook because black will lose his queen if he captures it. Don't be too concerned about this, you played a good move which kept an advantage and that is good enough as you are a new player trying to learn and improve. There was a little nuance in the position which you didn't see but as you improve the more details you will see. We're not gods like the engines, it's about playing consistently good moves without making big mistakes rather than playing the absolute best move every time.

Thanks a lot this explanation really helped me, I will make sure to think the moves ahead more carefully next time.