Question: What are reasons for someone to do Pilates as his or her main fitness activity (like how other people do strength training, yoga, or running)? What does doing Pilates entail?

Answer: 
The overall number 1 reason to choose ANY activity for fitness:

***You Enjoy It***  

Nothing is more important than finding an activity that you discover you like doing and are eager to come back to each week. You won't make a permanent positive change in your life otherwise.

Now, reasons you might choose to try out Pilates and see if it's that kind of activity for you:

1. Good posture has positive side effects in many other areas of your life. It affects people's impression when they see or meet you. It prevents discomfort and chronic injury from office work. There are plenty of full length books on the benefits of good posture.
2. Learning to habituate biomechanically correct movement improves your performance in every other athletic endeavour. Dedicated pilates study will make flawless form in a movement like the barbell squat feel like a totally normal natural thing you do. Watch some video of an elite strength athlete like Mariusz Pudzianowski competing, notice the absolute perfection of movements even while doing totally absurd things. Learn to move properly as a matter of habit, rather than something you focus on for certain specific exercises then go back to having turned in hips and rolled in shoulders the rest of the day.
3. The right selection of equipment can provide strength training in muscles that are so weak it's almost impossible to work them without having other muscles take over and compensate. If you're coming from being sedentary, you definitely have some muscle groups that are like this.
4. Likewise the right selection of equipment can provide plenty of resistance for strength training push and pull compound body movements in all planes. If your goal is primarily putting on muscle mass, would using a barbell be faster? Of course, but it's not the one and only option for getting stronger. A Pilates program will combine strength training with balance, posture, mobility, flexibility etc. (As opposed to a program where you might say "I move a barbell for strength then do these other accessory exercises to keep those other things in line.")
5. You can participate in progressive classes where you learn new things about moving and about your body as time goes on. Variety is important for some people, learning more, new, harder ways to say, do a lower body push movement, may (or may not) be more rewarding for you than the powerlifting approach of pursuing ever-greater weights in the same set of core movements.
6. It is difficult to explain and I don't have references to research, but you're just plain happier and more functional when your hips aren't tight and your sacrum is properly aligned. There are lots of theories on the effect such as clearing up "neuromotor static noise" that have various levels of research around them. I can just tell you anecdotally it has been true for me.

**Footnote 1:** As you indicated, I am referring to Pilates practice as the primary, not only exercise. I do personally also do the major compound body movements with dumbbell/barbell for additional strength work, but time-wise I put much more into Pilates than moving iron. It simply shifts the emphasis off of adding muscle and onto those other things. I am not saying that *only* Pilates does those things and barbell training does not, it's just, how do you want to balance where you put your effort among all these competing concerns? Do you want to get stronger and have your posture gradually improve as a side effect, or fix your posture and get gradually stronger as a side effect? I find more time on these aspects and less on straight strength works for me.

**Footnote 2:** I am of course referring to studying the Pilates method in an equipped studio with a good instructor. Not flopping around on the floor of the living in front of a DVD "toning your tummy" with 2-lb. neoprene dumbbells :)
The overall number 1 reason to choose ANY activity for fitness:

***It causes the desired adaptation in your body***  

Nothing is more important than finding an activity that causes the adaptations that move you toward your goals. You won't make a permanent positive change in your life otherwise.

There is no good reason a person to choose pilates as their *main fitness activity* (assuming the goal is fitness, rather than pleasure, or for a feeling of doing something useful).

It perhaps would be useful or pleasing as a supplementary activity, but *for a main activity*, strength training, cardiovascular conditioning, or a sport that works these attributes work would always be better alternatives to pilates.