The Wrench is most commonly used to unlock enemy vehicles, allowing to quickly get away with the stolen vehicle and/or use its inventory.
It is possible to use the Wrench to unlock your Faction's vehicles. Bear in mind that unlocking and/or stealing your Faction's vehicles is strongly frowned upon and should be only done with permission or when under urgent conditions, use with responsibility.
Although the Wrench can dismantle the vehicle lock, it does not undo the squad claim on a friendly vehicle. Entering the drivers seat of an enemy squad-claimed vehicle will remove the squad claim.
Vehicles can now only be wrenched by friendly players 5 minutes after the owner has exited it, reducing the chances of vehicles being stolen by griefing players. (changed since Update 0.22)
Equip the Wrench.
Crouch C near an object that can be dismantled / a vehicle door.
Press LMB to begin dismantling the object / lock, you will see a prompt and animation that you are dismantling.
Once dismantling is finished, the character will slightly raise again and the object will disappear / the vehicle will be unlocked.
The dismantling process takes about 5 - 10 seconds to finish, beware as you will be left vulnerable during this period!