Playing the exiled card follows the normal rules for playing that card. You must pay its cost, and you must follow all applicable timing rules. For example, if it's a creature card, you can cast it by paying its mana cost only during your main phase while the stack is empty.

If you don't play the card, it will remain exiled.

A creature has been dealt excess damage if Nahiri's Warcrafting deals more damage to it than the minimum amount of damage required to be lethal damage. In most cases, this means damage greater than the creature's toughness, but consider any damage that was already dealt to it that turn.

Unless an effect allows you to play additional lands that turn, you can play an exiled land card only if you haven't played a land yet that turn.

For a planeswalker, excess damage means damage in excess of the planeswalker's loyalty (the number of loyalty counters it has on it). For a battle, excess damage means damage in excess of the battle's defense (the number of defense counters it has on it).