To unlock the By Invitation Only trophy in Assassin's Creed III Remastered, you will need to be invited to join a club by completing an activity that corresponds to the club.
The list of activities required can be found below:
Boston Brawlers: Defeat one enemy with your fists.
Frontiersmen: Find a trading post in the Frontier.
Hunting Society: Kill and skin one animal.
Thieves: Pickpocket £100.