The irony in this statement concerns the way in which Nixon and his adminstration was infamous for lies and misdirection. To launder something normally means to make it clean, but the pun in this sentence gives it a new meaning that is perfectly in keeping with the various misdemeanours concerning the "truth" of Nixon's presidency.
This has to do with the difference between connotation and denotation of a word. The denotation, or definition, of the word launder has not changed. It still means to take something soiled and make it clean. The connotation, or meaning we give to the word, has changed. Before Nixon, the word launder brought to mind the idea of clean laundry. After Nixon, the connotation of the word launder changed to a more negative view. Since Nixon used the word to discuss his illegal and soiled financial activities, the word began to call up images of dark and illicit dealings rather than clean and bright connotations. I don't think the speaker of this quotes was against Nixon, but he was being ironic.
Launder is ironic because it has to do with cleaning money, but the money becomes dirty when it is cleaned, in other words illegal. So the term money laundering refers to hiding illegally earned money and making it seem legitimate. Nixon engaged in many illegal schemes, thus the reference to him.
Let's also consider the other polar opposites in Nixon's administration between actions and words. Leadership was violating the Constitution, democracy was cheating in an election, not bombing Cambodia was bombing Cambodia, and not being a "crook" was, well, being a crook. One could say Nixon was a master at saying one thing and doing or being another, which dovetails nicely wth Zinsser's quote and sentiment.
There is also the laundry-related pun of "Irony." If we don't take the word at face value, then to Iron something is to make it flat or remove the wrinkles. Laundering money would remove the "wrinkles" of criminal activity, thus ironing it.
Zinsser's chief purpose obviously seems to have been to define "irony," although he may also have been implying a personal political statement as well. Creative writers often employ irony to communicate means subtly, indirectly, and by implication, although irony is often misunderstood and sometimes is not even noticed at all. Zinsser's efforts both to explain and to illustrate irony are therefore useful.
He clearly said this because he was anti-Nixon. Obviously, the word "launder" has always been connected to things that are dirty. After all, there is no need to launder something if it is already clean.
So, I don't really see the irony here. It is somewhat ironic that a person who ought to have represented law and order (by his own claims) would end up giving us a new term for a kind of crime. However, I don't really see that using the word "launder" is ironic because it has always been used to taking something that is dirty and making it clean.
There's not really anything more to add. Up until that time, "launder" carried connotations of making items clean. Pres. Nixon and his associates used the word to describe their arrangements to "clean" money from any associations or history that might prove troublesome if not completely illegal.
It is indeed fitting irony that "launder" became a dirty word; as rather than "clean" it was used to cover up ones "dirty deeds." In this instance, President Nixon engaged in a number of illegal acts to prevent those involved in the Watergate scandal from being discovered; particularly that money paid to those involved appear to come from legitimate sources. It is now commonplace to speak of "laundering" money from shady or even illegal sources by attempting to hide the real source, an act which is in itself "dirty."