The tech and operations are HOW they scaled. But the demand and service quality are WHY they scaled.
100 years ago, anyone with a car could drive you across town for a fee. It was a dangerous job and, left unregulated, became difficult to provide safety, ensure supply/demand matching, etc. So cities stepped in and regulated. Taxi commissions became entrenched, powerful and politically mafia-esque. They focused on core areas where they could get the most business with certainty: airports and hotels. You could only drive if you had a city-issued medallion and cities set caps on the number they issued. Those who had medallions fought cities to prevent new medallions from being issued to reduce supply and increase demand.
Supply & demand grew increasingly imbalanced. And, as more young professionals entered cities with fewer parking spaces for cars, demand grew rapidly- but taxis did not serve this market as they focused on hotels and airports.
Uber entered via black cabs and then P2P. Neither of which required medallions, thus circumventing the taxi supply/demand constraint. Technology enabled them to overcome the trust and safety issues of previous jitney services, increase speed and reliability, and enabled them to scale to a new market: urbanites.
Growing parking constraints from increasing urbanization, increasing ubiquity of smartphones/apps, changing (younger, professional) demographics of cities, and deteriorization of both taxi and transit service quality have all supported the growth and demand for their service.
Note that their growth has not been universal across all markets. Even though they are in 300 cities, most of their demand comes from a handful of cities sharing certain similar characteristics for certain types of trips at certain times.