This is unlocked by collecting all 50 Monster Cards within a single save file.
That in turn requires you to collect all 50 Nuggets (used to trade for the cards of the 7 Sirens).
(I suggest getting the 2nd Siren's card first, as it gives you constant Magic recovery! This makes much of the rest of the game considerably easier sooner than it would have been.)
Some are fairly common, while others will likely require you to reload an area MANY times for the card to drop.
There are a few that are not obvious how to get:
-Crystal Crab (looks like a pile of walking Gems): You have to start Dancing to suck the Gems off them and make them vulnerable to attacks.
-Pincher (those invincible jumping... pinchers): You have to first get the Jellyfish and have that equipped so you can deal damage to Pinchers and get their card.
-Dragon Head (The fire-shooting statues): You have to use your Turtle Fusion to break them.
-Siren Man: These are not available until late in the game (post 5th boss).
-Siren Soldier: These are not available until late in the game (post 5th boss).
Advice: Get Max Attract ASAP. Also equip the Lobster Jogger once you have it, as it increases the Attract range even further. (Yes, you CAN just Dance to Attract items, but having it be passive is just great!)
Also, any unneeded duplicates can be sold to a merchant in Armor Town.
He will always leave you enough to equip their ability, no need to worry.
He will give you a maximum of 999 Gems. However, he always takes all the spare duplicates, so stockpiling for multiple times selling isn't an option.