Section I, Lines 16-20 Summary

Get out the microscope, because we’re going through this poem line-by-line.

who talked continuously seventy hours from park to pad to bar to Bellevue to museum to the Brooklyn Bridge,

Wow, they can talk for seventy hours straight!
They jump between very different locations, such as from Bellevue, a famous mental hospital, straight to a "museum."

lost battalion of platonic conversationalists jumping down the stoops off fire escapes off windowsills off Empire State out of the moon,

They are still talking. The speaker compares them to a military "battalion of platonic conversationalists," which means they talked a lot about Big Ideas. Plato was the Ancient Greek philosopher who came up with the theory of Ideas.
They also have some pretty impressive gymnastics skills. They jumped off everything from fire escapes to the Empire State Building. This, obviously, is an exaggeration.

yacketayakking screaming vomiting whispering facts and memories and anecdotes and eyeball kicks and shocks of hospitals and jails and wars,

Still talking, talking, talking.
They discussed everything from their cherished memories to the pain they suffered in "hospitals and jails and wars." These are people who have seen some rough times.
They also vomited between all the talking, screaming, and whispering. Or, maybe by "vomiting" Ginsberg means spewing out more words. The next line suggests this might be the case.

whole intellects disgorged in total recall for seven days and nights with brilliant eyes, meat for the Synagogue cast on the pavement,

By total recall the speaker means that they have access to all their memories, which they "disgorged," or vomited out in conversation for a week straight.

who vanished into nowhere Zen New Jersey leaving a trail of ambiguous picture postcards of Atlantic City Hall,

Now the speaker tells us that these people sometimes vanish entirely. They skipped town, and no one knows where they have gone. Three of the likeliest options are: 1) nowhere, 2) Zen, a Buddhist state of enlightenment, and 3) New Jersey.
They sent postcards from Atlantic City, New Jersey, which would seem to suggest option 3. But maybe those three names all stand for the same place. (Jersey is Zen?)