“I once sent a dozen of my friends a telegram saying 'flee at once - all is discovered.' They all left town immediately.”
It implies that his friends each had some terrible secret they were hiding. He didn't actually know what that secret was, but the cryptic telegram was enough to scare them into believing that their secret had been discovered.
I was able to find a number of references to this quote online, but none cited a source. So it's not clear to me where this quote came from or if Mark Twain ever actually said it, since Twain is frequently misattributed with dubious but humorous quotes.
This is one of those observations that is made in every generation, and attributed to a notable wit (who may indeed have put it into a concise form). The earliest printed version I have found is in Kipling's Puck of Pook's Hill, chapter 4.
'"Write to any man that all is betrayed," said De Aquila, "and even the Pope himself would sleep uneasily. Eh, Jehan? If one told thee all was betrayed, what wouldst thou do?"
'"I would run away," said Jehan. "it might be true."
That his twelve friends fled indicates that they all had something to hide. (That's the joke.)
Who said this phrase and when is in question. A a modest search could not put the words in Mark Twain's mouth at a given place and time. Some have attributed a similar quote to Sir Arthur Ignatius Conan Doyle. (He was 23 years younger than Mark Twain, the pen name of Samuel Clemens.)