Talent Tree is the place where you can invest the Talent Points that you have earned on the Battle Tower.
Each hexagon provides a specific buff or stat increase that will be applied to all of your Pokémon. Stats that are applied to all of your pokémon we usually call them Base Stats or Base Power.
Upgrading those Hexagons costs Talent Points but also costs Gold.
There are four trees that you can invest on, and each one is unlockable according to the amount of Talent Points that you have invested on the previous tree.
If you ever want to reset your trees, because you might have invested your points wrong for example, you can do it with the Reset Talent button. This will cost you 500 Diamonds, but you will get all your Talent Points back and 90% of the Gold you've spent.