I think there are really two parts to this question.
1) What inspired people to check out a ukulele for the first time?
2) Why do people like it so much after they try it?
I don't have much insight into #1 though it probably involves people like Jake Shimabukuro, Eddie Vedder, and mumford and sons.
But I think I undersand part 2. The ukulele is popular because it is versatile and relatively easy (perhaps 'accessible' is a better word than 'easy'). Versatile instruments include other instruments like piano and guitar. They are well suited to a variety of styles of music and sound great by themselves or with other instruments. Contrast this to less versatile instruments like flute, bass, or trombone. The problem with piano and guitar is that they require quite of bit of time and expertise to master. In fact, they require quite a lot of time just to play pretty basic tunes. Ukulele on there other hand is incredibly accessible. People can play easy songs (even popular tunes that they like) in the first week. And after that it is possible to progress quickly into finger picking and beyond. Please don't be offended when I say that ukulele is relatively easy. What Jake Shimabukuro and James Hill do is not anywhere close to easy. They are equally as talented and doing equally difficult things as anyone playing guitar or any other instrument at the highest level. I just mean to say that getting started and making progress at the beginning is easier than with some other instruments. On top of that they are typically cheaper, more portable, and smaller (guitars can be too big for children or those with smaller hands) than other versatile instruments. All of these things contribute to its accessibility.
Almost every one would like to make music. And ukuleles get you to the point of making music much faster.
There are certainly other reasons to like the ukulele (its happy sounding, people like hawaiian music, etc) but I think the versatility and accessibility are the major reasons. Those are certainly what drew me in.
The question I always have is why weren't they already popular? Why did they lose popularity in the past?
Anyway, that's what I think. I am sure others have different ideas and am interested to see what they say.