Because promotional material encourages players to go to them. The idea is that players will form playgroups around their local gaming store. In that sense, "Your" LGS would be the (hopefully) nearby store that you would to to to play games/purchase materials.
This is in contrast to other brick and mortar shops that you may go to for a particular event, but is not local, or convenient.
As to why we use the particular phrase LGS, that's a little harder to pin down. Most likely, that's just the wording that TSR decided to use for Dungeons and Dragons. Since Magic the Gathering (the first of the wave of trading card games) hails from the same distributor, it makes sense to use the same terminology. Other TCGs follow suit, and here we are.
Additionally, the D&D and MTG communities formed before online retailers became mainstream. There would have been no need to differentiate between a physical and online retailer. This would explain why the E-Cigarette community uses said phrase: Because E-Cig supplies have been widely available online from the beginning of the Craze (Or, at least, since it became mainstream).
Either way, Communities such as those around E-Cigs and Electronics do not share roots with the D&D/TCG communities (though there is definitely overlap), and so developed their own jargon. Plus, of course, you don't expect to play games at a store that sells Vaping Fluid.