You get card packs for completing certain parts in the story. You also gain packs for reaching certain thresholds by completing expedition challenges (different than the story ones, which are linked to a different achievement), missions, and earning medals.
"Missions" is a different game type of Expeditions along with Score Attack, Chapter Replay, and Chapter Replay Elite.
Also you can purchase them from the marketplace for in game credits or real money. DON'T WASTE YOUR REAL MONEY!!!
Getting a gold medal in score attack with earn you 10k credits you can use to buy the cheapest card pack twice. Keep buying the cheap pack until you earn 300 cards.
This popped for me finally after grinding out credits to keep buying packs. It finally unlocked it at 280 cards. I know I had sold at least 100 cards and used probably 20 in the score attack game mode which should have put me over the 300 cards a long time ago.
So... I would recommend not selling any cards until this achievement unlocks. I think using them would be fine and you still get credit but if you sell any those will more or less count against you. This is my experience anyway.
NOTE: Some cards can only be used in certain game types, Score Attack, Missions, Chapter Replay... again though, use with caution until the achievement unlocks to avoid excessive grinding to earn more cards than you have to.
NOTE: At one point, the game would not let me purchase any more cards due to me not having enough room. I am not sure if it was a server problem, glitch, or not being far enough into the story. If anyone else experiences this please comment.
NOTE: It has been brought to my attention that you will need to maintain connection to the Square Enix servers to keep track of cards, medals, and leaderboard results. You can play the Expeditions without the internet access but the achievements may not track and unlock accurately.