I'm a natural storyteller. In fact, I follow the advice, "Never let a few facts get in the way of a good story."
Frequently, music has stories attached. Think about Bizet's Carmen. That's one helluva story. Some pieces don't necessarily have stories but have characters attached (Flight of the Bumblebee, "The Butterfly," and so on). Some pieces are completely story-free (Mozart Concerti, for instance, or a Bach Sonata).
When I can, I read up on the stories and history of the pieces I'm preparing or teaching, the history of the composer, and when possible, the historical era and setting, political influences, fashion, gossip, and anything to add color to my image of the music. This tends to really bring the music to life for the students -- and it gives them a structure upon which they can hang musical ideas.
You see, kids don't have a lot of experience pulling stories 'out of thin air,' so to speak. Public education is not very tolerant of kids with too much independent thought and original ideas, and certainly doesn't encourage it.
Here are some examples of how I might talk a student through this process.
"Ok, you're playing the Habanera from Bizet's Carmen. Ever hear of the story? No? Well, long and short, Carmen's not a very nice gal. She's got a boyfriend -- a bullfighter (Toreador!). She works in a cigarette factory, and at lunchtime, the girls from the factory crowd around the second floor window to watch the English soldiers who march drills in the city square right there. They laugh and whistle and generally try to cause trouble. Carmen picks out one guy in particular, and decides she wants him to fall for her. She goes down there one day, and starts to flirt. That's the song you're playing. She's strutting around with a rose, singing (in the opera). He's oblivious, talking to his friends, drinking some water, cooling off. She tries harder. Then! She gets annoyed. That's where the key changes to major and gets snappier. That's Carmen saying, "Eh! I don' wan' chu! I don' NEED chu." all the while, she's still strutting around him. She's spiteful and angry, but can't quite give up. ...."
And so on. Depending on the age of the student, I may or may not go into the rest of the story (he falls for her, he's got a lovely English rose of a fiancee at home, pining for him, they run off together with the gypsies, her boyfriend gets mad. He kills the soldier, then he himself gets gored by a bull. She dies. The end. Blood and guts everywhere. The only one left standing is the English girl who is forever desolated by his subterfuge."
When they are actually playing the music, I'll play the piano part and sometimes I make up words to help them remember the attitude of the action. "I am Carmen, and I'm so Hottttt," etc. "I don' need CHU, I don' wan' CHU!" etc. It makes them laugh, but they remember how to interpret the music!
I've had students from years ago (20 or so) contact me and they STILL remember some of those stories.