in Dragonvault, to heal damage means to remove damage from a champion. Each champion has a toughness score which signifies the amount of damage that, once accumulated, would result in the champion's death. For example, Radin has 8 toughness. This means that if at any time, Radin accumulates 8 damage, he dies. Damage continues to accumulate unless it is healed, either by an ability from a champion like Celaine, a card like Healing Potion.
There are two ways to track damage in Dragonvault. The primary method is to use dice (preferably red d6), which are placed on the champion to show how much damage has been accumulated. If a champion is undamaged, it would not have any counters on it. But, for example, if Tom took 3 damage from a Bear Trap encounter, you would place a red d6 on him, with the 3 visible. That way, the rest of the players at the table know that Tom has accumulated 3 damage. Tom's toughness is 7, so if accumulates 4 more damage, Tom dies.
The second method to track damage is to use dice which correspond to the champion's toughness (or close to it), for example a d8, a d10 or a d12, and treat toughness like hit points. The die would start at the toughness value, and then work down as the champion takes damage. For example, Astrid has 8 toughness. So the player might use a d8, and set it on Astrid, with the 8 visible to the table. If Astrid took 3 damage from a Hired Muscle, the player controlling her would move the d8 value from 8 to 5. If Astrid was healed by a Food card, the player would change the d8 value from 5 back to 8. To some players, this method comes more naturally, and it is perfectly acceptable to do it this way. Just be sure and discuss with the players which method is going to be used beforehand, to avoid confusion. One danger of using this method is that some players may subconsciously be tempted to move the die value higher than their toughness score. Remember that a champion cannot heal "beyond" their toughness, since toughness is not like hit points, but rather a number denoting how much damage accumulated would be fatal to the champion. A subtle difference, but one noting nonetheless.