Easy to pickup and play. Other genres of game are more technical (fighting games have advanced stick/pad motions + button combinations, each RPG has a completely different battle system, abilities list & items to learn, sports games have advanced control features in game that make regulars WAY more advantaged to win vs casuals).
Shooters are walk, look, shoot, throw frag, aim down sight, change weapon. 8 year olds pick up FPS games really fast. Have that 8 years old kid try doing that with Diablo 3, Madden or Tekken Tag Tournament. Poor kid will get frustrated just trying to learn the basics!
Also, guns have always had a machismo appeal to them as well.
I think the ease of accessibility is a major attraction point for me. At its base, a shooter game is the REALLY simple process of moving a crosshair over a target, and pressing a button at just the right time. And the result on the screen is often pretty satisfying.
It also caters to a lot of interests like in the power fantasy where an action hero solves all of her or his problems by shooting things.
And shooting things can be fun! Whether it's a concrete wall, an exploding barrel, or a guard with a tranquilizer, there's a certain satisfaction that comes with precisely shooting a target and seeing it explode, break apart, give off some smoke, or fall asleep because of your action.
As a result of this shooter mechanic being so basic, a lot of games are designed around this concept, so shooter games are very prevalent and popular as a result.
I guess shooter games are popular for the same reason why movies with violence are popular.
Mostly because online multiplayer exploded. They were always pretty big, even dating back to the days of Doom and Duke Nukem, but the online thing turned the entire genre into a casual shoot-fest. This is one of the reasons why so-called "real" gamers tend to look down their noses at FPS fans.
I can't stand multiplayer so I miss the days when we used to get great shooter campaigns, like the earlier days of Medal of Honor. Hell, I thought the original Red Faction on PS2 was awesome, and I loved Rage. I just can't stand running around in circles doing the same thing over and over and OVER. Which is all online multiplayer is, as far as I'm concerned.
I love some shooters like Battlefield and Halo, I even like Bioshock & Fall out because they are genuinely fun online and offline experiences. However nothing compares to the Witcher 3: Wild Hunt - the best game I have ever played.
Exciting. Good storyline, good action. Doing things that would be really cool in real life, so we can pretend and mentally get in the story.
Also, with xbox live, you are in fact fighting and competing against other live players, so you're not just fighting a computer program, but you are "killing" or being "killed" by other people. Competition.