Comes from Dutch word “bezig”.
You can usually uncover stuff like this by googling “etymology word” where word is the term you’re interested in, and “googling” is the search engine of your choice.
I also heartily recommend “RobWords” on YouTube. He frequently delves into the mysteries of how English adopts and then mangles words from other languages.
Because saying it like "B-uh-zy" sounds like "buzzy". Saying it like "B-oo-zy" makes it sound like a horny way of saying "bussy".
Becuase the written language isn't the definition, it's just the record of the spoken language.
Spelling is an attempt to reflect the sound, but within the constraints of the rules of the written form. This is why the pronunciation key has its own structure and format.
Also history and etymology. The origin is Germanic besig https://www.etymonline.com/word/busy