There's a few things, I think, that contribute:
Word of mouth, a la social media, is definitely up there. So many people hear what a great experience it is, and excitement grows.
The fact that it sells out faster every year encourages people to sign up even quicker the following year.
Disney marketing has, over the past few years, definitely positioned it's events as newbie friendly, almost to a fault in my view. (In many cases, it encourages people to start running, and adopt a healthier lifestyle. For others, it merely encourages doing just the minimum--or less-- needed to get the medal. I don't want to digress too far, though.)
Running as a hobby is somewhat in vogue lately, and running a distance race has become a bucket list item for a lot of people. (See previous comment about the split in how that works out long term.) Disney events happen to be a great place for that, given the "fun" atmosphere.
Multi-day challenges (about which I have mixed feelings as well) are another factor. The added "challenge" or the draw of a special medal gets many registering for multiple events who otherwise might only have been interested in one of them. Though the multiple events might give the illusion of more slots available for registration, there are not necessarily more overall number of participants.
I suspect this isn't really the same as what happens in some other events (Boston and NY come to mind), which are more popular for their world-class status and long-standing history.
Regarding the size of the shorter races, I concur largely with what's already been stated, but wanted to amplify it a bit. If you've done a half or full with enough people (Disney or not), you've probably noticed that congestion tends to decrease as you progress through the course, with the first mile being the worst, and so on. For a short event like a 5k, there just isn't enough race distance for the crowd to ever thin out. While adding lots of corrals with staggered start times is possible from a technical standpoint, there are pragmatic reasons why Disney probably doesn't want to do it. It's one thing to have corral launches last 30-40 minutes in an event that has a completion window of 3.5 or 7 hours; that 30 minutes isn't all that significant. But, for an event that goes from start to finish in 45 minutes or less, a lengthy corral/wave process would (proportionally) have a much greater logistical effect for the shorter events.