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Steam's Rich Presence is what populates all the details about a user in the Friends list and importantly for multiplayer games, it's what enables the "Join Game" button on the Steam friends list.
The Join Game button located on the Steam friends list in the Steam client lets a user simply click a button and it will launch the game if it's not already running and provide the game with connection details so it may join that player to their friends game.
What happens when a user presses that button is
To enable the Join Game button on the Steam Friends list simply set the "connect" field of the user's rich presence.
This will set the local user's connect information to whatever string you provide for connectionData. What exactly you should pass into as your connection data depends on how your game handles the connection.
This would usually be the CSteamID of the server you are playing on, or the IP and Port if it's using a traditional TCP/UDP connection or the CSteamID of the "Host" if you are playing P2P.
Simply clear the data