"The man from the West, his egotism enlarged by success, was beginning to outline the history of his career." What does this sentence tells you about Bob's personality?

The quote you mentioned—especially the words "his egotism enlarged by success"—tells us much about Bob's personality and character. Those words highlight Bob's materialistic predilections. Essentially, Bob's sense of self is predicated upon his ability to gain material success. In that light, he's extremely proud of the fact that he excelled in the competitive business world of the West. His pride in his accomplishments is clear, and he can't wait to tell his old friend Jimmy Wells all about it.
When he meets the man he thinks is Jimmy, Bob excitedly "outlines the history of his career." He thinks that he has bested Jimmy in life, and he can hardly hold himself back from openly boasting about his success. Bob is so preoccupied with his telling that he almost fails to realize that the man he is speaking to isn't Jimmy.
When he finally snaps "You're not Jimmy Wells," it's too late. The plainclothes police officer tells Bob that he has been under arrest for ten minutes. So, the quote basically tells us this: Bob's primary purpose in life is to gain material success. His ego is predicated on his ability to outperform his business competitors, and he enjoys boasting about his exploits after the fact.