Now a question:
If they didn't explain the dance and the story behind it before they danced, would you have any idea what story the dance was trying to tell?

It depends on the choreography and how well the dancers interpret the choreo and the music.

Melissa and Ade's Tyce Contemporary dance? Absolutely understood the story purely from how they danced. There was emotion, frustration, anger, fear, vulnerability and at the end, hope. Didn't need the explanation.

Contrast that with Melissa and Ade's cha cha cha. That was just steps.

I think sometimes the choreographers tell a story in order to get you in the right frame of mind for the dance but a really well danced program evokes emotion in the watcher regardless. So you can attach that emotion to a specific story, or you can tell a story through dance and then relate what you feel to a more concrete example, using words, and say, yeah, it's kinda like that.