On page 168 of the core rulebook, it states that stun grenades have 'disorient 3'. The only other section of the book that I find disorient in details its effects of the target gaining 1 disadvantage die, but no details are given as to the mechanics. Does a stun grenade automatically cause disorient? What does the 3 mean?

First, I would highly suggest reading the section in the book that covers weapon qualities, beginning on page 154.

Disorient 3 means that the effects of the Disorient last for 3 rounds (in this case, a setback die added to checks for three rounds).

The Disorient is not automatic; it requires 2 Advantage to 'activate' on a successful attack. (This is the general rule for most weapon qualities, unless otherwise specified in their description.)

One thing to add is a weapon quality requires a successful attack roll, that does not mean inflict wounds past soak. So you can Disorient targets, Ensnare, etc without actually exceeding soak. You only need to exceed soak to cause Wounds, obviously, and inflict critical hits.