You are considered "strapped" when you are in possesion of a fire-arm. But it started off as a term to describe when you have a Mac 10 or some other semimachine gun or uzi on a strap hanging from your shoulder under your clothes.
Strap predates a submachine gun hanging from a strap. Virtually any concealed weapon's holster requires it to be "strapped on", hence the term "to be strapped." Any shoulder, leg, or ankle holster is strapped on. Furthermore, nobody wears or wore their Mac 10 or Uzi hanging from a strap on their shoulder.
A guy wearing a jockstrap is said to be strapped, or strapping.
Straps were required under our shorts for gym at the high school I went to. The gym teachers would do random strap checks - by having us pull down our shorts or sweatpants just enough to show we were strappin'. Any unstrapped boys would have to sit out class, and get detention.