A main requirement appears to be that these organizations are underdogs. The pattern would not emerge in post acquisition PayPal for example, or something like Google or Facebook.
It's not just smart people. Bell Labs had really smart people but they were not powerful.
While they are underdogs, they are also well funded. They are not starving startups, but mid sized organizations. This gives everyone room to grow and plenty of challenges.
Chemistry among the talent is also unusually strong. Leaders have to be open minded, willing to make risky moves.
Another sign is that the members are usually willing to give up much better offers for what they have. People very rarely quit the group, and usually these are the kind of people who regularly get offers from top companies. It's a black hole - it sucks smart people from all over.
And finally, there needs to be some bigger bad out there that unites them to focus. This could be something like a IPO. It could be a much larger competitor. Or it could be them against the world.
This pressures them to stay united, work together against a common threat. Usually this is the missing link that prevents any regular group of smart people from becoming a mafia. This pressure builds camaraderie and creates unusual growth.
First you need to find something to work on and then work on it. That's how you get experience.
You might list your skills and interests and ask if there is anyone here who has an idea you can invest in by working on it with them.