The museum is a building to show off how awesome your team is. It contains a trophy room and shows a history graph of your team's league and cup results.
The museum also gives a small weekly income.
There is no limit to the level of the museum, so you can always upgrade it. There is a ranking of highest level museums in the game.
When a manager leaves his club, the level of the museum is divided by two.
You need to have a popularity of at least 500000 before you can build the museum.