You are obsessed with using and maintaining a wide variety of conventional firearms. With each rank of the Gun Nut perk, you gain an additional 5 points to the Small Guns and Repairskills. With all three ranks, the player gains a +15 bonus to Small Guns and Repair.
The Gun Nut perk grants access to more effective gun mods.
For each rank taken in the Gun Nut perk tree, the Sole Survivor will gain access to a better tier of gun modifications. Gun Nut can be taken four times, each time granting access to an additional rank of gun modifications (ie, Rank 1 of the perk accesses Rank 1 modifications, Rank 2 of the perk accesses Rank 2 modifications, and so on).
Having this perk will allow your player character to say they have heard of Reilly's Rangers before, to which Reilly will reply "You're pretty sharp, kid."
When rescuing Initiate Pek having this will allow you to fix his gun (even though you can do it by having a mediocre Small Guns or Repair skill as well).
The perk image shows a missile launcher among the collection of weapons. The missile launcher is actually a Big Gun, so does not get a bonus from this perk.