It costs one action and allows the hero to perform an attack (with a melee weapon only) and then add +1 heart to the roll. It's really not that complicated, and I fear for the future of your group if this is all it takes to start a huge rules arguement. No offense intended.
If it helps you to visualize this, think of it like this: There's the Attack Action and then there's "making an attack." The Attack Action is an action which any hero or monster may perform (at least once) during their respective turns. "Making an attack" is to follow the sequence of steps described in the rulebook under "Combat" on page 12.
The Attack action allows any figure capable of performing it to "make an attack." Certain skill cards and monster abilities also allow the concerned figure to "make an attack," usually with a few extra bonuses or caveats. These skills and abilities, however, do not cost an action unless they say they do on their respective cards, and that would be a "Use a Skill" action, if it applies. (Note that the vast majority of skills and abilities that allow a figure to make an attack also require an action. Go figure.)
Also note that the rules for monsters explicitly state that any ability which allows the monster to "make an attack" also counts as that figure's one attack for the turn.