What happens next?
The number-one ingredient for a story is the tension of an unsolved mystery. Stories set up a questions and delay answering them. The simplest example is a question in the first sentence with the answer delayed until the second sentence:
"You know who Bob's favorite singer is? Meatloaf!"
That's not a very interesting story, I know, but compare it to this:
"Bob's favorite singer is Meatloaf."
The first version evokes (just a little) tension. The second doesn't.
Now imagine telling the first version but walking out of the room after the first sentence:
"You know who Bob's favorite singer is? ----- "
That agony is what you should strive for. Because the most basic human urge that makes us want to listen to stories is the need to know what happens next.
Interesting how much that relates to the concept of punchlines. "Always end with the funny part" is one of those simple/obvious rules that I wind up needing to constantly re-remind myself of when writing new bits.
There's also a bit on curiosity:
Curiosity is the juggernaut that drives storytelling.
If you immediately tell us what happens next -- or if there is no next ("Bob's favorite singer is Meatloaf") -- then there's no hook.
Practice this simple question-delay-answer structure over and over, in all your communications. I mean in emails, text-messages, Quora posts, and so on. You're not going to become a good storyteller by learning how to go into storytelling mode. Instead, turn yourself into someone who tells stories all the time. Make stories a natural part of the way you communicate.
I don't mean you should start emails with "Once upon a time..." I mean you should always be aware of posing a question, pausing, and then answering.
"You bet I'll come to your party tonight, and I'm going to bring something tasty! My grandma's snickerdoodles!"
"Practice this in all your communications" is good advice. Whatever you want to do onstage, you should do it a little bit all the time. That way, you're not trying to go from zero to 60 the second you hit the stage.
There's more good info at the rest of that piece. Also related to this stuff: Patton Oswalt on pointing fingers and building tension.