Unlock the Museum by donating 15 creatures to Tom Nook
Talk to Blathers after the Museum opens
Donate at least one of each:
Fish
Bug
Fossil
Sea creature
Wait until Blathers mentions wanting to expand the Museum
Find Brewster on a mystery island via Kapp’n boat tours
Talk to Brewster on the island
Return to Blathers and tell him about Brewster
Wait for the Museum café to be built
Visit the Museum café to find Brewster inside