The reference is to the "break-in" process. The word process is left out.
A mechanical system, at times, needs a break-in process because the mechanical properties smoothen themselves into place. A typical example is shoes. Shoes need to be broken-in so that they conform to wearer's feet. Another example is new car engines. Cars should not be driven above a particular speed until their engines are broken in. A third example would be the brakes in a car. With those both "broken-in" and "bed-in" are used.
For the gun break-in process, the reference is in particular to the barrel break-in process, although it also applies to the rest of the gun assembly. The gun break-in is also referred to as "seasoning your gun" or "seasoning your barrel". The process is arduous, requiring the user to fire anywhere from 100 - 500 shots from the gun and cleaning the barrel intermittently, anywhere from every 20 to every 5 rounds fired. Also, the process, if performed incorrectly, can damage the gun.
The gun manufacturers basically do this process on new guns beforehand, so that the gun is ready for normal use.
It is both a time- and cost-saving selling point, thus the advertisement.