He will look after me and make sure I get through this situation well, with a positive outcome.

"Martin Eden," by Jack London! "He'll see me through" means "He'll take care of me." The "through" part means "until the problem or danger is over." Suppose you call an ambulance for someone who is hurt. The ambulance arrives. Suppose that you don't leave. Suppose you ride with them to the hospital and stay with them until a doctor has seen them. You have "seen them through." The person in the passage feels that he's in danger, in social danger. He's worried about doing the wrong thing and making a bad impression. He is about to meet people who are wealthier and better educated than he is. He in fact makes a social mistake by putting his cap into his coat pocket, and the other person corrects the problem. I really like "Martin Eden." I think it's my favorite of all Jack London's books, and I hope you enjoy it. It is NOT in easy English. Keep in mind that it was written in 1909, and writing styles were different then. It is hardly a secret that the book is loosely autobiographical, and echoes aspects of the real Jack London's life.