Learn English – What’s the meaning of “for the best”

In the context of a phone call, a person who says "take me off speaker" wants to say something that only the other person should hear and not anyone who might be within earshot of the phone's loud-speaker. It means "turn off the phone's speaker and hold the phone to your ear". The pronoun "me" there refers to himself as the identify of the person whose voice is coming out of the phone's speaker on the other end.

If you understand all of this already, then your question is not about language but about the movie.

To me, it sounds like a variation of to come from money (M-W):

Definition of come from money
: to have a rich family • His wife comes from money.

The writer is saying that Matthew's family is very rich. It does not refer to Matthew and his wife (the couple). It refers to his parents and his lineage.

To answer your other question (in the edit): no, it's contrasting people's perception of Matthew with his actual history. The author says that they come across as (= appear to be) a "well-heeled, successful couple". This suggests that they became wealthy on their own, without help. But, the truth is that Matthew's family was rich. The author is suggesting that Matthew did not become wealthy on his own.