Does anyone know how one gets to perform at Carnegie Hall?

Practice. Practice. Practice.

Or you could just be extremely rich. (see Florence Foster Jenkins)

You or your manager would contact Carnegie Hall, and if the price and the act are right, there you go.

You’ll need an agent (or manager, as @iphigeneia says) and they arrange all the stuff.

I am just guessing that you have to already be established in entertainment as a noted act.

The staff picks the acts under the guidance of the artistic director.

It appears their website has a good amount of information on hiring one of the halls: http://www.carnegiehall.org/Rentals/Hall-Rentals/

I think that would be the easiest way to get there, if you can afford it.

@Hawaii_Jake stole my answer!

^^ Ditto.

My chorale was first invited to sing there with three other groups. On the basis of that, we were invited to do our own performance of the Verdi Requiem as part of a series of Carnegie Hall concerts. How it works for others, I don’t know. An agent is involved.

Agents, a nice profile as far as experience goes, and a few audition pieces that you’ve recorded. They get tons of these inquiries a day, so you have to stand out, have patience, and be able to meet their quality standards. It’s worth it, though.