Starts Powered means that an Ability has enough Energy to be used immediately. After the Ability has been used, the Energy must recharge in order for the Ability to be used again. This is the so-called Energy Cost value. It's a cooldown value. In the game this value is visible in a tooltip that appears when you press and hold an ability icon.
Since the first Ability is always fully charged at the beginning of a battle, Starts Powered only affects the second and third Ability of the hero. Also, it is only relevant for the attack team, because in a Raid the defenders have all three abilities fully charged.
The opposite can also happen. If the Hero is hit by an attack with Energy Loss, the ability may not be able to be used immediately, but has to recharge over a few rounds or has to be actively recharged.
For more details see Energy Loss, Energy Gain and Extra Energy.
Energy Drain does not affect the currently available Energy, but increases the recharge time of the abilities.
The following heroes that have an ability that Starts Powered:
Drop Dead Gorgon (Special Attack): Ranged Spirit attack with a chance to Stone the target, purging all Buffs and Debuffs on them and turning them into a statue, or kill them instantly. The chance of killing the target increases by 10% for each stack of Venom purged on the target. Starts Powered. (Golems and Constructs are immune to being Stoned)
Solar Shot (Special Attack): Armor piercing Ranged Physical attack for [92.5% ATK] base damage that Blinds the target (lowering their attack and defense, 73%@lvl 80) and reduces the target's energy by 1. Starts Powered.
Into the Pit (Special Attack): Buffs team +ATK and Banishes target enemy, purging all Buffs and rendering them inactive for 2 turns, then returning them to the dungeon at 50% of their current HP (Epic Bosses cannot be Banished). Banished enemies count as being defeated when no other enemies are active. Starts Powered.
Crystallized Carapace (Special Attack): Taunts, gains 75% Damage Reduction, and heals allies the damage taken. Enemies who hit him have their attack lowered by % (55% @ lvl 80) while the taunt is active. Starts Powered.
The Calm Before (Special Attack): Silences all enemies and Buffs allies' Damage. This Hero gains stacks of Storm Power equal to the number of Light or Honorbound Heroes on the team. Starts Powered. (The Empowered Buff to applied to all allies and the caster to double (2x) their damage.)
Ground Pound (Special Attack): Taunts all enemies and buffs +DEF. Allies take reduced damage for the duration of the effect. Melee attackers become Swamped. The next attack on a Swamped target heals the attacker. (not a debuff, can't be purged.) Starts Powered. Snow Day: Consume Snow Day to buff allies +DEF.
From the Ashes (Trait): Passive with counters as cooldown: The Phoenix stores power over 3 turns. When fully charged, Solaris [and all beast allies] resurrect at the end of the round. [Starts Powered.]; This is a ranked trait [I/II]. Description in [square brackets] is for rank II, 'From the Ashes-II'.
Chroma Shift (Special Attack): Ranged Physical attack on target, with a chance to apply Venom. 2x damage if target already has Venom. This Hero then shifts to target’s Element, cleanses all debuffs from self, and gains 20% DEF and Max Health until the end of the next round. Cleanses self again when shift expires. Starts powered, but cannot be used while shifted. Cooldown: 6 Turns.