Hear him out and he’ll tell you about his past and the Enclave, and more importantly, how those two go together. You’ll need to do most of these things to get Arcade talking to you about this quest, but if you wait until the later stages of the quests For the Republic, Part 2, The House Always Wins, or Wild Card: Change in Management, he’ll offer you this quest with fewer requirements. Talk to the remnants, tell them of the reunion, then listen to Arcade aftewards for some backstory (right).
Earn Arcade Gannon’s trust and he’ll ask you to help gather the remnants (left).