They did not break the taxi laws of every city they operated in. The taxi laws of many cities only required a "medallion" for taxi-style car services that allowed passengers to "hail" the car on the street.

So-called "black" cars where passengers arranged transportation in advance were governed differently, and required no licensing or a different and more permissive kind of licensing. Black car services were already widely available in many cities from lots of different companies, often independent owner/operators - it's just that no one had provided a service that worked across many different cities and was available conveniently as an app.

I won't claim that Uber and Lyft broke no laws, because there are a lot of jurisdictions in the world, and I don't know the laws of all of them, but many cities regular taxies and black cars very differently.

Edit: Why the downvotes? Am I wrong? Evidence would be appreciated.