What exactly do these startup mentors mean when they say 'Tell a good story'?
They mean that pitches about software applications are effective when they focus on:
Telling a story is a real block for people, and my best advice is to keep it simple. Don't go depth-first and come up with fictional characters with motivations, family history, etc. Go breadth-first -- walk through product usage, brainstorm ideas at each stage and try to put them together and make them flow. Remember that you're really just looking for a couple of great little bits to string together and go for minimalism.
For most apps, you want to keep that breadth-first style. Something like, "Now that I've uploaded my photos I can email the album to my group, I can tag faces or I can create a slideshow. Let's create a slideshow..."
Here's an easy structure to start with:
Now, if you do want to tell a longer story, like a screencast, you should have one or more pieces of storyline conflict, what screenwriters called "reversals." Have a story where your user:
Anything else which shows that your app is exerting some control of the storyline is a good thing, because it shows that the app is actively affecting the user's experience.
And please watch the famous Ira Glass video about the basics of journalistic story (anecdote plus insight). He's a master. Then become inspired to come up with an anecdote to match the insight of "this never could have happened without the user using my product."
It's simple and hugely important.
Take a step back. Think - what is the main point I want to get across? The "take away" I want my audience to walk away with.
Then how do you get there in a logical way, basically melding together the pieces in a way where one leads to the next.
So for example:
You get the point. You tell a story that all flows.
Simple: Beginning. Middle. End.
If you tell any idea in that format it's a story. Start simple and write two different endings. Know the story like the back of your hand, then tell it to people. See how they react. Ask them what they liked, didn't like -- and what their favorite story IS.
Repeat the process until you have the effect you want.