Someone figured out the right dimensions and multitouch programming, with enough CPU to play games.
They abandoned the handwriting recognition that was important for business users and students, and went with stuff everyone can do -- stuff that most people need in their lives. Entertainment, mostly. Games. Art -- fingerpainting and music. Video and music. Books. New instruments, new ways to play, old games in your pocket or your purse/briefcase. Clever little things that only such a device could enable... you want to do it on your GPS, but that's so single-purpose designed, and hard to manipulate...
And then it can do all that GPS and personal-management stuff, too, that companies tried to sell smartphones and PIMs on.
To be fair, they do sell smartphones on these features, too. Smartphones can do all this same stuff. But there are situations where it's nice to have a bigger screen than a smartphone's while not carrying around something as big as a laptop. My experience is that tablets of the iPad size are good mostly for leisure activities around the house: surfing the web, reading e-books, watching video, playing games.
(DIsclosure: I used to work in smartphones and am now in tablets.)
It's basically all of the reasons people have smartphones, but it's big enough that some things that are not totally convenient on smartphones (reading ebooks, showing pictures to people, browsing the web) are a lot easier. Add a wireless keyboard and a stand and you have something like a modular laptop -- handy for writing email or posting to livejournal or whatever, and then when you're done you can put the keyboard away and keep Internetting away, or playing Angry Birds, or whatever, which is convenient in a restaurant or other location where desktop space is in short supply.
Because it's solid state and not as bulky as a laptop it's not as fragile either, so I can do things with it that I would hesitate to do with a laptop, like bringing it into the kitchen so I can read a recipe from it.
I don't think there are single areas where it's a lot more convenient than either a laptop or a smartphone, but there are a ton of areas where it's slightly more convenient, and they add up.
A tablet isn't as unreadable as a smartphone and it's not as heavy as a laptop. That's the attraction.
I've never even *touched* an iPad or other tablet. But I wish I had one. Why? Because I've looked at smartphones and e-readers and can't read ANYTHING on those fuckers. And I've owned a laptop; it was good, but not convenient.
I guess I can see times where I'd use the laptop if it weren't so heavy and fragile (!) so I don't. And recently acquiring a Blackberry, and trying to surf the web (or even read lengthy emails) - ugh. So thanks for the insight. Maybe I understand now.
Functionally, a modern tablet is really a big smartphone. So it's any situation in which you'd think "I'd want to do this on my smartphone, except I'd like it to be bigger" (and don't care about the corresponding loss of pocket-portability).