What does “cash out” mean in the context of “aides who have flamed out, lashed out or cashed out.”?

In a casino, "cashing out" is when you take whatever chips you have left to the cashier cage and exchange it back into cash, ostensibly done for the evening. Or, in an informal situation, taking your chips off the table and changing them back into cash - either way, effectively & voluntarily retiring from playing any further. Not to mention cutting your losses or preventing further damage to your financial status.

In a political sense, "cashing out" seems to imply trying to get out of a situation, or escape while they're ahead and on good terms as opposed to being fired.

Usage and meaning of “cash out” in 2017-18

According to Merriam-Webster, the phrasal verb cash out first appeared in print in 1971. In the US, where it more commonly used, it carries positive connotations because it signifies a gambling person who is either collecting their winnings or leaving the table while they still have some money left.

However, in the context of White House staffers, who have either resigned or been publically fired under the Trump administration, this note of positivity seems to have soured. To cash out, in the context of the article, refers to someone exchanging knowledge, experience, or any commodity for cash. In the New York Times article, the author suggests that President Trump's ex-employees are "milking" their tenure at the White House for all it's worth. (See, What's the evolution of the phrase "milk it for all its worth"? for more details on its meaning.)

The following expressions used in the same article, flame out, and lash out have strong negative connotations. Flame out would refer to those staffers whose employment at the White House were spectacularly brief: Sally Yates, former attorney general, was dismissed after 10 days; Michael Flynn, the former national security advisor, was fired after 23 days; and Anthony Scaramucci's official tenure lasted just six days.

Meanwhile, lash, as defined by M-W is “to make a verbal attack or retort —usually used with out” The NYT was most likely thinking of Anthony Scaramucci, whose intemperate words and crude language became infamous. Another former White House assistant, Sebastian Gorka, is also notorious for his fierce verbal attacks on "fake news".

Thus, the author of the article used cashed out as a rather neat rhetorical device. I'm not absolutely sure whether flamed out, lashed out, or cashed out is strictly a tricolon device, but according to this source, a tricolon is a “series of three words, phrases or sentences that are parallel in structure, length and/or rhythm”

How common is “cash out” in American English?

Very.

In what form does this "cash out" take? Usually, its book deals, TV show appearances or public speaking fees. Former White House staffer, Omarosa Manigault, whose contract was terminated on December 13, 2017, is currently appearing on CBS's Celebrity Big Brother, which is scheduled to end on February 27.

Omarosa Continues to Cash Out on Her Time in the White House Because That’s What Soulless People Do
The Root

Former political figures tend to make some extra money speaking after their tenures end, and Omarosa is no exception. The former White House aide signed with a speakers bureau on Monday and will begin asking for up to $50,000 a speech, according to The Daily Mail. source

Reince Priebus, former Chief of Staff, who was replaced by Gen. John F. Kelly, "retired" from the White House administration on July 27, 2017

Priebus in talks to join lucrative speaking circuit

• Another former White House official is seeking to cash out. President Donald Trump’s former chief of staff, Reince Priebus, has been interviewing with speakers bureaus around Washington, D.C., as he seeks a place on the lucrative paid speaking circuit, according to multiple sources familiar with his life-after-Trump plans. (source)

And another former employee of the White House, ex-FBI director, James Comey, is said to have been offered a lucrative "tell-all" book deal

Former FBI chief James Comey signs $2m book deal

• James Comey, the former director of the FBI who was sacked by President Donald Trump in May, has signed a deal for a book, ostensibly on leadership and decision-making, that is scheduled to come out in spring next year. Financial Times

Cash out literally means:

To dispose of a long-held asset for profit: ‘Hard-pressed farmers are tempted to cash out by selling their valuable land. (AHD)

In a figurative sense, it is used suggested a radical change in your professional and personal life:

cash out - choose a simpler life style after questioning personal and career satisfaction goals;

• "After 3 decades in politics, she cashed out and moved to Polynesia"

Assuming that the meaning of the phrase is clear, let's talk about the motivation for phrasing it exactly as they did, which I assume are phonological and semantic.

First, there is a mellifluous aspect to the prosody. Three trochees in sequence. You sometimes hear of the rule of three in English rhetoric, where it's preferred to list things in threes rather than twos or fours. I don't know the psychological basis of it, but it's very common in English rhetoric to list things in threes.

Next, of course, is the rhyme and alliteration of lash and cash. Alliteration has a long pedigree in English poetry, and you find it as the top poetic device in Old and Middle English works.

Finally, there is some semantic symbolism going on by using three -out compounds in a row. There are two types of idiomatic expressions in English that are compounds formed by adding -out to a verb. The first changes the meaning in a mostly idiosyncratic way, except that -out usually suggests some sort of altered state of reality or consciousness. E.g., space out, smoke out, pass out. Then there are other expressions formed with -out where out usually adds a meaning of getting expended: flame out, die out, run out. Mixing the two types of expressions together provides an underlying meaning to readers: the Trump White House staff are fickle, unstable, ephemeral.