Share intrigue with a player who previously shared intrigue with you.
That is what the achievement says. But how exactly do you gain it? I think I missed a feature here.
Intrigue is the name for the events when the AI pops up and tells you something such as "Elizabeth is plotting against you!" or "Bismark is planning a surprise attack against someone!" or "Atilla is building The Oracle!"
You gain intrigue when you have a spy placed in an enemy city. When your spy uncovers some intrigue you'll get a popup, and if there is a subject identified you can click on the popup to go to the AI interaction screen with that AI and you can share the intrigue. You can also share intrigue later on by going to interact with the AI on your own (you don't have to follow the popup).
If you miss the pop ups, the Espionage overview screen has an intrigue tab which lists all the pieces of intrigue you've had throughout the game. It also says what turn you got the info, so you can tell whether it's still relevant.
Primarily, intrigue is a great way to improve diplomatic relations with AI by telling them when someone else is plotting against them.
So to get the achievement, share some of your intrigue with an AI who has shared intrigue with you previously.