I believe it is within my power to articulate the appeal of Instagram.
1. It's ubiquitous and easy.
It's simple and it's easy and it works. Everyone's using it and all your friends are already there. For a lot of people I imagine Instagram is simply the default. I spent a few minutes browsing through the people friends were following, seeing what people are doing with it and I ran across quite a few who had signed up, taken one or two pictures and abandoned it. A lot like Twitter. The app is well-designed and enjoyable to use and the easiest/best/simplest/fastest way I've tried to publish photos.
2. It's really easy to make anything look good.
Toy camera/Polaroid/etc. emulators are like the color equivalent of shooting black and white film. It's a shortcut. Most any idiot with a basic grasp of photography can take decentish photos. I mean, not everyone, I've seen plenty of terrible Instagram pics, but you still have to do a LITTLE of the work, but you can basically ignore lighting, color, contrast, focus, etc. Just pick a filter that makes it look ok. Not to mention the whole aesthetic of it is so hot right now. But that's a different/longer conversation.
3. It's limiting.
Limitations on creativity are freeing. This is the biggest point for me. I can lose hours on one shot in Photoshop exploring virtually unlimited options. I'm sure anyone who's ever done anything can relate to option paralysis, in any medium. With Instagram you take a picture, crop, turn brightness on or off, pick a filter and you're done. Example. I took this at the end of a car wash night before last.
It's nothing amazing, but it was online by the time I was rolling out the wash. Sure I could have done it with any number of apps and photo hosts, but it wouldn't have been as streamlined and it might look even worse. Or possibly better.