When an opponent's monster declares an attack: You take damage equal to the attacking monster's ATK, and if you do, banish that monster. During the End Phase of your opponent's next turn, Special Summon that monster to your side of the field.
You do not immediately get your opponent's monster.
You essentially take a direct attack from the monster.
Tips
You will need to take the full damage to steal your opponent's monster. If for some reason the damage is negated or reduced, you will not be able to banish and steal your opponent's monster.
Heal lots of LP
In decks like Aromage or Fairies, you can find yourself in situations where you have tons of LP but nothing to really do with it. In this case Memory of an Adversary could be useful for gaining advantage and building a board presense. Ideally you are able to steal a powerful monster or even your opponent's boss monster.
You can use cards like Mirror Wall or Wall of Disruption to lower the opponent's attack so that you take less damage. Since you did take the full damage your opponent's monster while it was on the field, the monster will be banished and will still be special summoned to your field with it's debuffs removed.
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