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What you do is, you mark down the rank and suit of a card near the element that allowed you to draw it (so an element of the mission you approached, or one of your own relationships you revealed). When you play a card take some big decisive action or reveal something, you need to narrate how the element connected to that card was involved in you taking that action or finding out.  

So there isn't any general requirement, say "you can only use a hearts card for an action that centers around seduction", or anything like that. But there's a specific requirement, focused on your game and what you already have introduced: the cards require you to reincorporate elements that have already been discussed at the table. 

So, say you have 2 cards in your hand: one is the 8 of hearts, one is the 6 of spade. You drew the 8 of hearts when you approached the prison your busted agent (also one of the characters, the Casualty) was being held in, by bribing one of the guards. You drew the 6 of spades when you revealed that you and the Planner, your mission head, have been lovers in the past. 

Now you want to do something big, let's say, check if the Casualty's claim that he escaped on his own from the prison is true, or if he's lying. 

To do so you have to use one of your cards. When you play it, you need to incorporate the related element: so you may say how you talked to the bribed guard, and play the 8 of hearts; or you may invent something about how your relationship with the Planner helped you in this, and play the 6 of spades. 

This helps making your story a bit tighter (instead of generating a lot of leads and only actually playing out a few), and also helps creating the tangled knots of relationships and rivalries you often find in the stories we're trying to emulate. It is, effectively, a labeling system - but a deliberate one. I don't quite see that as not using the cards, but ymmv of course :)

We also use the 4 Aces to quickly identify the operation critical elements on the chart, so all players remember them. But all dynamics and interactions are exactly the same; simply, the game pre-determines what card you'll obtain if you do an approach action centered on one of those elements.