The lesson that the Bramble teaches the Fir-Tree is that there is nothing wrong with being a Bramble. He tells the Fir-Tree that there is good in not being a Fir-Tree. He teaches him that he is fine just the way he is.
My answer related to the story because the Fir-Tree was trying to give the Bramble reasons to not like himself and want to have grown up a Fir-Tree.