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The Stage Manager was Dead to Begin With
And he's using the Mental Time Travel powers apparent with death to tell the audience the story. The characters never really interact with him unless he's standing in for another character, which Emily shows us is possible in Act 3. This would most likely be much after the events of the play, allowing for the Professor and Mr. Webb's contributions in Act 1.
Alternatively, he's this afterlife's spirit guide of some sort.