In 13th Age the escalation die adds to your chance to hit and also causes certain abilities and monster tricks to "pop" when the die reaches a certain point. You can achieve similar results in 5E by using the following ideas:
Add the Esclation Die to Damage: apply it to damage once on your round instead of to hit, since damage/hit points are open-ended in 5E but accuracy isn't.
Modify Recharge Rates for Monster Abilities: Use Escalation Dice with power-recharges for monsters, using one of two methods: either roll a D4 for the recharge # but add the escalation die each round, or roll recharges as normal, but when the escalation die reaches a recharge # the power automatically is restored.
Grant Inspiration: Every character gets an "Escalation die Inspiration roll" at the start of combat: when initiative is rolled, everyone rolls a D6. When the escalation die hits that # they gain inspiration.
Restore Legendary Ability Points: Legendary creatures regain one legendary point on escalation die 6.
Regain Abilities: Adventurers regain one spell slot or ability of choice (i.e. action surge or any other depleted resource that requires a short or long rest to regain) on escalation die 6.
The goal of the Escalation Die is to keep combat moving and to "speed it up" if its taking longer than normal. The secondary (and cooler) effect of the Escalation Die is to make combat more epic and over-the-top. Try it out and see!