Pip makes his request of Biddy after his first visit with Estella and Miss Havisham. This is significant, because it is his visit with those two very strange characters that is the catalyst for Pip's disappointment and embarrassment of his own lowly station in life. Estella had commented on his rough working hands, and had been disdainful that he was an uneducated poor kid, instead of a gentleman. For the first time in his life, he was ashamed of his status, where he came from, his family, and his station. So, he decides to go to Biddy and become educated. In their society at that time, education was the key to changing one's circumstances. Pip decided to do what he could to change his life, even if he couldn't leave, or change his circumstances. He would better himself, to make himself worthy of Estella's love and esteem.
Biddy predicts that once he becomes a gentleman, and a learned man, he will reject his family and friends and former life, and consider himself too good to associate with them. Pip shuns that prediction, but time will only tell if she was right. I hope that those thoughts helped; good luck!