Obviously, they do the fundamentals well. Good brand. Good experience. Good word of mouth. Good PR. Etc. Etc.
But after my interview with Ryan Graves, the head of Global Operations at Uber (https://www.growthhacker.tv/ryan-graves), it became clear that they are operationally advanced and this is a huge part of their success. I'll explain.
Uber isn't just a single startup, it's essentially dozens of startups rolled into one because every time they enter a new city they have to establish themselves from essentially nothing (except whatever brand equity has reached the city ahead of them). This means finding/training drivers, marketing to consumers, and building out local staff to manage operations for that city.
This is where Ryan Graves comes in. He has a protocol of everything that must be done, and in what order, and by who, to ensure the best chance of success in a new city.
So how has Uber grown so fast? Essentially, they figured out how to grow in one locale and were relentless about refining their launch process to recreate that initial success over and over in new cities. No plan works for every city, and they've had to adapt in many situations, but it is still a driving factor for their success.