After donating five creatures to Tom Nook, you can bring Blathers to your island and build the Museum.
Donate Five Creatures to Tom Nook
After you learn how to do DIY, you can get a fishing rod and net. Use them to catch bugs and fish. Talk to Tom Nook to donate five creatures, and he will give you the Blathers Tent Marker.
Build the Blathers Tent Marker
Find a place on your island where you want the museum to be, and build the Blathers Tent Marker there.
Donate 15 Creatures to Blathers
The next day, you can start donating bugs and fish to Blathers. After you have donated 15 creatures, Blathers will say that the museum can be built. The museum will replace Blathers's tent the following day.