Energy Refills are items that, like what the name implies, recharges your Energy. There are two types of Refills: Mission Energy, and Alliance Mission Energy.
The Mission Energy Refill can be bought for 40 
.
Alliance Mission Energy Refills are only available to buy on Alliance days, and the first costs 30 , then 45, 60, 75, and 90, limit 5 each day.
Both of these items can be obtained as prizes for completing certain Objectives, or in the case of the Alliance Mission Energy Refill, the Alliance Crystals and the Raid Store. The Mission Energy Refill completely refills your Mission Energy, while Alliance Mission Energy refills only recharge 1. If you don't use the refills, Mission Energy takes 6 minutes to recharge, and Alliance Misison energy 60 45 minutes.
It isn't wise to use your refill when you have at least 3 Mission energy left over; you're wasting the energy and in return, you lose three extra "advances". Use it when you have no energy left, or if you can't travel any further if the mission requires more than 1 energy to keep exploring.