Conducting elections is a traditional and exclusive government function.
Such is deemed as a traditional and exclusive public function.
Anything that produces the equivalent of a state-ran election is a state action subject to the incorporated provision of the Constitution.
From this case: the Jaybird Democratic Association, a private unregulated political club, hosted a pre-primary election that excluded black voters.
Historically, the winner of the Jaybird primary ran unopposed in the real state primary.
Because the Jaybird election was a quasi-primary election—a traditional and exclusive government function—it could be regulated under the 15th Amendment.