The reasons are legion and depend, to a great extent, on each individual asking the question.  However, the basics can all be boiled down to what I call the Three E's of Ease of Use, Economics, and Effectiveness.

Beginners frequently have problems taming recoil.  This can lead to bad habits such as flinching, which makes accuracy almost impossible to achieve.  More importantly, a shooter who is recoil sensitive might get discouraged trying to learn with more punishing rounds and give up shooting altogether.  With the light recoil of the 9mm, inexperienced shooters can learn the basics of sight picture, trigger control, etc., without being battered by the gun.

Of course, with experience, most shooters learn to handle recoil properly and can move on to other calibers if they choose.  However, recoil force has a direct impact on things like muzzle flip, which in turn affect how quickly and accurately a person can make follow-up shots.  This shouldn't surprise anyone.  The more the muzzle flips up, the more time and effort it takes to bring it back down and on target.  Time equals marksmanship, and the more time you have to take your aimed shot, the more accurate you will be.  Therefore, the 9x19 allows a shooter of any given skill level to be faster and more accurate when firing multiple rounds.

Most tactical handgun trainers agree that the ability to put multiple rounds downrange into a target is critical for self-defense.  The obvious conclusion, then, is that the 9mm gives the shooter an edge when performing double taps (two rapid shots to Center of Mass, or "COM") and similar defensive techniques.

By comparison, most other defensive ammo calibers are significantly more expensive.  Discussing just practice ammo (like FMJ or Blazer), the 9mm is usually two to four dollars cheaper per box than comparable .40S&W and .45ACP ammo; 10mm and 357SIG ammunition can be twice as expensive!  Of course,  shooters of these other calibers can save money by reloading, but that requires (1) an outlay of substantial funds to buy the reloading equipment and (2) time and effort spent sorting brass and loading ammunition.  Many shooters would rather not be bothered with all of that, myself included.

Also, in my experience, great deals (like specials, sales, etc.) on 9mm ammo are much more common than for other calibers.  Surplus NATO 9mm "ball" ammo makes an excellent training/practice round.

Less expensive ammunition, of course, means more ammunition.  Whether you want to spend $10 or $100 or $1000 each month practicing, you'll get more for your money with 9mm.  More ammo means more practice, and more practice means greater skill.

In a defensive shooting situation, shot placement is much more important than tiny differences in so-called "stopping power."  The only way to improve shot placement is by practicing.  By switching to 9mm from .40S&W or .45ACP, you can practice half again as much for the same cost.

When considered along with the 9mm's inherently reduced recoil, the economic efficiency of shooting 9mm means that you get better, faster, cheaper.

I won't lie to you.  They're right.  The average 9mm load probably isn't as effective as a defensive round as the average .40S&W, .45ACP, 357SIG, or 10mm round.

Whoa, hold on a minute!  Did he just say the 9mm isn't as good as those others?

No.  I said that the average load wasn't as good.  When you start to look at the best loads in each caliber, you begin to see that they're almost identical in terminal performance (ability to penetrate, expand, and otherwise wound a violent threat).

Here are some samples of performance in bare gelatin:

As you can see, the 9mm versions of most "premium" loads are very close and sometimes superior to the .40S&W and .45ACP versions.   It's all about bullet design, not bullet weight or velocity.

The problem is that while there are few "bad" loads in the other calibers, there are tons of "bad" 9mm defensive choices out there.  Many rounds either fail to expand or fail to penetrate, or both.  It is important that you, as a shooter, do a little research and choose 9x19mm ammunition which is tailored to your particular needs. 

So for 9mm, load selection becomes paramount.  (Click here to see CALIBERS recommendations in 9x19mm) But once you choose a good load, it works just like a good load in .40S&W, .45ACP, or any of those other calibers.   Sure, it's not as heavy as the heavy bullets, and it's not as fast as the fastest bullets.  But if it penetrates the same, expands the same, and disrupts tissue the same, who cares?  All else being equal, I'd prefer a cheap, easy to control gun rather than one that makes me work harder and spend more money to get the same results.

Oddly enough, I haven't found a single person so far who is so unimpressed with the stopping power of a 9mm that he is willing to stand downrange and catch one fired out of my Beretta.  8-)

Stay safe ...