The network effects are pretty obvious actually: as the number of people on the platform increases, the amount of time a driver has to drive to pick you up becomes increasingly shorter, thus enabling the drivers to spend more of their time with a passenger in their car (which is when they are earning money). If drivers are making more money, Uber can pay them less (reduce the subsidy).
Old taxi cab companies can't compete because they have to drive much farther on average for each pickup