How is flyblackbird.com operating under Part 91K?
Long story short, Blackbird appears to be set up so that the customer leases an aircraft and separately hires a pilot to fly it. That makes it a 'straight' part 91 operation. However, the FAA has stated that pilots using Blackbird are holding out and require part 119/135 certificates (with some technical exceptions, such as 91K).



Blackbird's terms and conditions for pilots specifically mention 91 B but not K and include a few interesting points (emphasis mine):


  
  Pilot services are offered as an independent contractor to users under a dry lease
  The pilot is not directly employed by the owner of the airplane
  The pilot operating the flight cannot own the aircraft being rented
  Pilot hiring is at the discretion of the passenger; BlackBird will facilitate matching pilots with aircraft based on passenger
  selected routes, pilot qualifications, and location and availability
  of both pilot and aircraft
  Pilot understands that passenger (lessee) renting the aircraft and hiring the pilot accept and maintain operational control of the flight; these operational duties may be delegated to the pilot in command.
  


In other words, the passenger rents an aircraft from an owner and then separately hires a pilot to fly it, according to the passenger's instructions. Blackbird is saying that they only provide a marketplace, and the customers actually lease and operate the aircraft themselves.

This is from their general term and conditions:


  The Platform provides a marketplace that enables users to (a) connect
  with and lease aircraft from third parties that own, lease, or
  otherwise control aircraft (“Aircraft Owners”) and to find and hire
  flight crews (“Flight Crews”), or (b) join flights with third-party
  operators or other BlackBird users on the Platform (“Third-Party
  Operators”).


And:


  If you wish to lease an aircraft from an Aircraft Owner and / or hire
  qualified Flight Crew, then you can use the Platform to search for
  available aircraft and Flight Crew


Because the owner just rents out the aircraft without a pilot, they aren't operating an air carrier. Because the pilot doesn't provide the aircraft, they aren't holding out. That means the only pilot qualification needed is a commercial certificate, although Blackbird also requires 500hrs, an instrument rating and a background check.

Now, whether or not that business model will survive if the FAA gets interested in it is another question. There was some discussion on the AOPA forums (members only) a while ago about Blackbird and I think the consensus came down to "technically legal, but pushing their luck with the FAA and DOT".

A semi-useful comparison here might be Uber. They've always claimed that they aren't a transportation company but rather a marketplace that just connects independent drivers with customers. That argument has succeeded in some places but not in others, and many countries and cities have decided that Uber is a transportation company and started regulating them. I have no idea whether or not that will happen with Blackbird.



Update: on December 17th, 2019 the FAA made its opinion very clear in a letter to Blackbird's lawyers:


  The information that BlackBird has presented leads us to conclude that
  the pilots participating in BlackBird's platform and using its app are
  holding out and thus are engaged in common carriage.


If you're a pilot who flew via Blackbird, their closing statement isn't reassuring:


  Accordingly, please expect further investigative activity into
  BlackBird's operations, particularly regarding its pilot database. In
  addition, we would be interested in learning of any action you intend
  to take in view of the jeopardy facing pilots who participate in
  BlackBird' s service.


There's a lot more discussion online, especially in Reddit (/r/flying) if you want to read more.