The Mechanophiliac badge is for "Using your Rig 100 times"
Rigs are your active ability, there are 4 total in the game (Stun, Blind, X-ray Vision and Heal). To use them you just have to activate them when available. Rig rechargers will help you refresh the cooldown.
There are some perks (prescriptions) and rig upgrades that speed up the recovery time, so just pick one you like and you'll likely get this after a dozen or so trials.
Also two handy things to note:
1. The in-game badge menu tracks your progress.
2. Leaving a trial after gaining progress still counts towards it.
Video guide for reference: