Talking to Trainer Arrken unlocks the required requisition. Fulfill "An Offering for the Arena," which consists of 2 Silverite and 2 Everite, at a Requisition Table. Return to Trainer Arrken to turn in the offering, then go down to the arena gate to begin the trials.
Note: Once you kill the last enemy in a wave, the next one begins in a few seconds. Cool-down times on skills are NOT reset; they persist between waves. Also, once you "kill" an enemy, they become immune to any further damage, their icon turns friendly, and their healthbar turns green. Your auto-attack will not immediately re-target on another still-hostile enemy.