The damage tokens are the dark blue/purple ones with a skull icon. The number on the token indicates how much damage it will do directly to the enemy’s health.
The key to this is to get your opponent’s health down low enough for this token to then finish them off. The trickiest part is not actually killing them with your attacks!
To calculate how many lives you will take from them in a given move just add your attack and any boost token equipped and take away their defence. Assuming they don’t have a boost token active (most early levels don’t use boost tokens) this will be the number of lives they will lose when you attack. Try to use this and time it so that you leave them on 1 or 2 health (depending on the strength of your damage token) and then use the damage token to finish them off.