For example, a couple of days ago I took an Uber. The driver that picked me up was a student. I asked him why he was driving, and he said, "Oh, well I came [to this city] to hang out with my friends, but they can't do anything for another hour. So I decided to do a few rides while I wait."
Uber was able to take the unused potential of someone just messing around in a city and use them to pick me up.
They are able to do that because they can watch everything that happens via the app. If someone tries to screw you over and take a long route if you're new to the city, Uber gives you a refund and the driver doesn't get paid. If the driver is driving like a maniac on bald tires while they talk on their phone, you can give them a low rating. Enough of those and they get kicked off of the service entirely. In other words, instead of an expensive and burdensome network of licensing and regulations, Uber was able to better regulate what was happening using technology.
The first time I ever took an Uber a driver came and picked me up, opened the door for me, gave me a water bottle and some candy, turned on the music I like, and was very safe and professional. And the whole thing cost less than a taxi. I haven't been in a taxi since.
So the app is the main attraction, but not because you can hail a car (though that's nice, especially when you're in underserved areas). It's because the app allows Uber to create a completely different economy around the supply and demand of ridesharing.