The initial phrase "First of all" acts adverbially on the whole sentence. It is the equivalent of putting a "1" in front of it:
1. My name is Amanda, but people call me Mandy.
It is therefore quite helpful to the reader if a comma follows it, to mark it off.
(In spoken English, you would pause after "all", with a distinctive intonation: this serves the same purpose as the comma in written English.)
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