"Weapon Spread" is how the shot from the weapon spreads out as it travels.
Basically you can think of it as a cone with it's apex at the muzzle of the gun and base at the extreme range of the weapon.
This can apply to weapons that fire multiple projectiles in one go - such as a shotgun or weapons that fire multiple projectiles sequentially - such as a minigun. Most of the projectiles will be in the centre of the cone, but a proportion will be at the extremes. This proportion could be constant or randomised (to a degree) at run-time.
This means that if you are closer to your target your shot will deal more damage as more of the projectiles will hit. It also means you don't have to be as accurate with your shots as a slightly off target shot will still deal some damage.
This also applies (to a certain extent) to the Pyro's flamethrower.