According to those involved*, that's what happened.
*who survived
According to the first DA who investigated:
"It appears Travis McMichael, Greg McMichael, and Bryan William were following, in pursuit burglary suspect with solid firsthand probable cause, in their neighborhood, and asking telling him to stop. It appears their intent was to stop and hold this criminal suspect until law enforcement arrived. Under Georgia Law this is perfectly legal"
Obviously, the deceased doesn't get to tell his side of the story.
DA George Barnhill's letter is here:
https://www.scribd.com/document/460352099/FullAhmaud-Arbery-case-Waycross-District-Attorney-explains-in-letter-why-no-one-was-arrested-in-25-year-old-s-shooting-death#fullscreen&from_embed
If it was perfectly legal under Georgia law, why were they charged and arrested?
I read the letter. A more self serving letter I have never read. “Here are the conclusions me and my guys came up with. You and your staff after investigating should reach the same conclusions.”