The Ener cost for your opponent's spell cancelled by Anti-Spell is still paid. When your opponent uses a spell, your opponent pays the cost of the spell. After your opponent pays the cost, you may activate Anti-Spell as a Spell Cut-In before the Spell's effect resolves.
Anti-Spell cannot be used in response to Life Burst abilities. However, Anti-Spell can be used against any spell that is used normally, even ones with Life Burst.