Before they built Basecamp, they were a well respected consultancy; they earned that respect with the great work they did. Overtime they kept growing their audience with their Signals vs. Noise blog which made it easier to get the initial traction for Basecamp and their other products.
Open sourcing Ruby on Rails also played a critical role in where they are today.
37signals no longer exists - they've gone "all in" on Basecamp, including rebranding the company to match.
The founding team started out, not by developing websites, but with a manifesto of doing business differently:
http://37signals.com/manifesto
Then they went forward and built their business. You can read a bit more about their earlier days here:
http://www.oreillynet.com/pub/a/network/2005/03/10/basecamp.html