Grover, or the "G-Man" is Percy's best friend in this series of books by Rick Riordan. He's an unlikely hero, because in his "human" life, he's bullied and misunderstood, cries easily, is physically inept--actually, crippled might be a better word (except when the lunchroom is serving a meal he likes, at which time he ignores the muscular disease which makes mere walking painful, and gets into line rather quickly). Grover is kind, perceptive, and a tree hugger of sorts, which is consistent with his heritage as a satyr. Getting a "searcher's license" will be necessary if Grover wishes to search for Pan, the god of wild places, as he tries to redeem himself after failing to keep Percy safe, as required by his role as a Keeper.
He wants to search for Pan and fulfill the dream of many other satyrs before him.
In The Lightning Thief, Grover wants a searcher's license because he wants to search for the missing god, Pan, who was known as the god of the wild. He wants to find him so that he can restore nature as the wild life in the world is diminishing as humans destroy it to build buildings and cities. It's just about every satyrs' dream to find Pan.
He wants to look for Pan, which would help to bring nature to its original state, to its best.
Grover wants a searchers license to get access to search for the God Pan and restore 'the wild', Pan is also a satyr just like grover so their it is probably a 'never leave one behind' issue. Its grovers dream just like all satyrs to find Pan.
Gorver wants to find the god Pan and restore nature to it's fullest.