one meaning of the noun 'run' is:
a continuous spell of a particular situation or condition : He's had a run of bad luck.
• a continuous series of performances : The play had a long run on Broadway.
So - it means the show had been written but not produced; and now its writer has found financial backing to have it staged and presented in a theatre, where it "finally gets a run".
(Prior to this, there may have been one-off performances of very limited productions (no fancy sets and costumes, no famous actors) but 'run' means a continuous presentation until attendance drops off and it is no longer profitable to continue to perform the show (or some other reason)