You can't get this kind of traction without sticky... really sticky tires.
The tires on the car above are B. F. Goodrich R1's (a Michelin brand) which were made specifically for Solo (or Autocross) events. Regretably, they've stopped manufacturing them.
My next tires (coming soon) use an equivalent R25 rubber compound, but are made by the Hoosier Tire Company. They still make sticky tires.