It's a common expression which means "I don't think I'm in a position to decide what things are - I'll just accept them."

Peachey is correct as to the meaning of the expression "who am I to judge?", but there is more to the transcript. The transcript is a play on words, it is a joke. First person says the honorific "Mister?", expecting to get the response of the second person's surname, eg "Jones". However, second person corrects the honorific to "Doctor". The first person interprets this as the surname expected and says "Mister Doctor" in the same way as "Mister Jones". The second person then corrects this with "It's Strange" indicating that his surname is not Doctor but Strange. The punchline is that the first person then interprets this as "it is strange", not the name, but the concept, responding with the phrase having the meaning as given by Peachey. THe joke is in the repeated misinterpretation by each party.

Peachy gave you the answer, but there is also a sort of pun or joke involved. Notice "Who am I to judge?" would often occur in a context like "I don't like body piercings myself, but who am I to judge?" or "It's strange, but who am I to judge?" Person X: Mr... [Pauses, expecting Dr. Strange to say his name]. Dr. Strange: Doctor. [I'm a doctor, and I want to be called "Doctor," not "Mister."] Person X: [Misunderstands, thinks "Doctor" is a surname.] Mr. Doctor? Dr. Strange: [Annoyed]. It's Strange. ["Strange" with a capital S. He means, "My surname isn't 'Doctor,' my surname is 'Strange.'"] Person X: [Misunderstands. Thinks Dr. Strange has said "It's strange."]. Maybe. Who am I to judge? I don't know the Marvel universe and I can't tell whether person X really misunderstands, or whether he's pretending to misunderstand in order to annoy Dr. Strange.