Imagine telling a story without narrative and without description; a story with no 'telling', no explanations of a character's speech or thoughts, why they said what they said, did what they did. No room for authorial voice, the writerly ego. No plugging the gaps in your story, or justifications for unconvincing character actions.
In truth there are theatrical devices that allow for 'telling' and for revealing character thoughts (soliloquy, chorus, monologue) but these are used less frequently in modern theatre pieces, and even if the playwright allows himself these indulgences, the bulk of the writing is pure dialogue.
So what's my point? In fact, I have several:
Narrative without narrative
The two meanings of 'narrative' – a story, the 'structure of the incidents' and the process of narrating the story - are useful when marking the distinction between story (or narrative) revealed through scenes and story (or narrative) revealed through narrative.
Writing for the stage teaches you how to tell a story through scenes. Aristotle's phrase to describe plot - 'the structure of the incidents' – is useful here as it highlights that the order in which scenes are presented becomes essential to the telling of story in the theatre. It's not just what happens next, but why it happens, each scene must be a necessary and natural consequence of the ones that preceed it.
Just imagine how much more powerful your story could be if it could stand up by its scenes alone. Just imagine how much more transferable to film.
The effectiveness of the 3 Act structure
Three acts equate to a beginning, a middle and an end. There are good reasons why plays are broken into acts – they prevent meandering storylines, sagging 'middles' and provide narrative drive and ascending dramatic tension. Acts are so good that Shakespeare uses 5 of them in his plays. All of them. Further discussions on the 3 Act structure here and here.
Less room for unconvincing characters
In the theatre every character in your story is going to undergo the kind of analysis and scrutiny that only a bunch of people with each of them focusing on one character can give you.
You give an actor a script and she's going to see the story through the eyes of that character and that character alone. She's going to ask you questions like 'so what's my motivation?' and say things like 'I don't think my character would do this.' Even the spear-carrier needs to understand and believe his character.
So all your major characters must have convincing arcs and believable actions, your minor characters must be consistent and whole.
Looking at your story from the angle of each and every character as an actor playing that character would can only improve your story.
Dialogue isn't just stuff people say
When it's the only technique available to you the dialogue has to advance the story and reveal character and be expositional and be convincing and fluent. Sound hard? Yes, of course it's hard, but how great will your dialogue be if it achieves all of these things.
I once attended Robert McKee's Story Seminar in London and he said something with regard to this that has always stayed with me. He was addressing when a voice-over can be justified in a film and his test was – if the film can stand up without the voice-over, then the voice-over is justified. The point being that the voice-over, the 'narrative voice', isn't making up for flaws in plot or character, but in fact adding value to an already great story.