Stormbreath Dragon

Creature — Dragon

Flying, haste, protection from white

: Monstrosity 3. (If this creature isn't monstrous, put three +1/+1 counters on it and it becomes monstrous.)

When this creature becomes monstrous, it deals damage to each opponent equal to the number of cards in that player's hand.

For each opponent, count the number of cards in that player's hand when the last ability resolves to determine how much damage Stormbreath Dragon does to them.
Once a creature becomes monstrous, it can't become monstrous again. If the creature is already monstrous when the monstrosity ability resolves, nothing happens.
Monstrous isn't an ability that a creature has. It's just something true about that creature. If the creature stops being a creature or loses its abilities, it will continue to be monstrous.
An ability that triggers when a creature becomes monstrous won't trigger if that creature isn't on the battlefield when its monstrosity ability resolves.