The attack value of a Hero is used as the base value for each hit done by the Troops aligned to the element of the Hero. The actual damage caused, depends largely on the defense value of the opponent, but there are many other variables at play as well.
For example, hitting two enemies at once divides the damage evenly for both of them. Having multiple heroes of the same element increases damage done by that element. And naturally, the elemental modifiers also affect the damage by a large degree.