I'm guessing this is about this, "The new law would require Airbnb and other short-term rental websites to post registration numbers on listings or email the number and name of the host to the Office of Short-Term Rentals, The City’s agency tasked with enforcing the regulations." (From http://www.sfexaminer.com/sf-poised-require-airbnb-list-regi...)
So "their" "free speech rights" are "violated" because they have to post registration numbers.
Basically, the city wants to hold Airbnb responsible for user-generated content. The city also wants Airbnb to hand over users' personal information. Both the EFF and the Center for Democracy & Technology think that these requirements violate federal law. (Though EPIC thinks otherwise.)
Airbnb gets paid for the rentals, not for the listings. Also, they actively control the content (try putting an URL in the listing). Mistaking Airbnb for a publishing medium for user-generated content seems disingenuous at best.
If they're actively controlling user generated content, then it's obvious they benefit from better content.