I can't think of any reason not to just carry a shotgun as a secondary weapon with Overkill. Seems like you'd want your attachment slot for something else. Was it just added to make a long list of attachments or do some players love this attachment?
To carry shotgun as secondary weapon you need the Overkill perk (shotguns are primary weapons in MW3).
Assault rifles are great weapons for mid to long range combat, but in close range fall behind SMGs and shotguns. Shotgun attachment gives you upper hand in close combat. However, note, that attachment itself is weaker then "pure" shotgun. Still it's quite handy when you are sweeping a building or you are in any situation with very limited space.
The real benefit from having a shotgun attachment is the versatility. It pairs very well with guns that typically don't do good at close range like 3 shot burst m16's or type 95s. This allows a user to balance some of the weaknesses of a gun with some of its strengths. A shotgun attachment is much faster then switching to your secondary (and you can't pick up a shotgun as a secondary without overkill anymore!). While not many people run with this attachment it has its niche for sure.