Paul Johnson is certainly correct in identifying the phrase "in my arrogant opinion" as being an ironic reversal of the set phrase "in my humble opinion." As this Ngram chart for the years 1600–2008 illustrates, however, the phrase "in my humble opinion" has been around for a lot longer than the Internet has:
In fact, the phrase attained its greatest popularity in written English between 1700 and 1850. But the earliest confirmed instances that a Google Books search turns up are even older, from speeches by a Major John Wildman, in "London's Liberties: or, A Learned Argument of Law and Reason" (1650, republished with some alterations in 1682):
I pray my Lord observe these words in this Record, the whole Commonalty that is to say the more able and discreet men of every ward. And to confirm this, if there be any need of it, we can produce another Record in 113 fol. libro C. Where election is said to be made by the Commonalty summoned thereunto : yet in Pag. 112, of the same, it is said men of every Ward did chuse : whence I collect that by the expression of the Commonalty sumoned hereto, is understood the twelve men from the Wards; so that it appeareth clearly in my humble opinion, that it was the Practice of the City for near two hundred Years, to Chuse by their Representatives, before it came to be the Usage of the City, to Chuse by the Livery-men of the Companies.
Now my Lord I humbly offer it to this Honourable Court, whether this opinion of the Judges about Elections produced by Mr. Maynard as the pillar whereon they build the lawfulness of the Liveries Elections, do not rather speak them to be unlawful, in my humble opinion, this that those learned Gentlemen flourished like Goliath's sword against us, slays themselves.
The first Google Books author to see the charm of "in my arrogant opinion" may have been Louis Leland, A Personal Kiwi-Yankee Dictionary (1980):
In my opinion change will not come about from total refusal to deal with south Africa but rather a combination of the carrot and the stick. 'We will play if you meet our terms,' rather than 'we won't play until you change your whole society'. Evolution not revolution is, in my arrogant opinion, the way to go.
Becky Symens, Acronyms Dictionary for Texting - Chatting - E-mail (2010) has this simple entry for IMAO:
IMAO In my arrogant opinion
Symens also has entries for IMBO ("In my biased opinion"), IMCDO ("In my conceited dogmatic opinion"), IMCO ("In my considered opinion"), IMHO ("In my humble opinion"), IMNERHO ("In my not even remotely humble opinion"), IMNSHO ("In my not so humble opinion"), IMNSVHO ("In my not so very humble opinion"), IMO ("In my opinion"), IMOBO ("In my own biased opinion"), and IMVHO ("In my very humble opinion"). The "Acronyms" appendix to R. Scott Perry, The Modem Dictionary (1994) has the same entry for IMAO:
IMAO In My Arrogant Opinion
thereby establishing that people have been using this initialism for more than twenty years.
With regard to what impression a writer tries to convey by using IMAO, I think it is a kind of playful acknowledgment that the accompanying assertion is an opinion, and that the writer thinks the IMHO formulation is disingenuous and (more often than not) insincere. But that doesn't mean that IMAO is used to emphasize that the writer is arrogant; it's more lighthearted than that, and as I noted earlier it plays on the reversal of the set phrase abbreviated IMHO.
As for when it is appropriate to use IMAO, I would say you should reserve it for situations where you are confident that your readers will not misinterpret your intentions. If you mean it jocularly, don't use it in a room full of humorless literalists who may take it as an admission of arrogance on your part.
Out of context, I wouldn't take it as an unfriendly phrase, although in some situations it might be intended aggressively (for example if it appeared immediately after someone else used IMHO in a comment with which the IMAO writer disagreed. But as is so often the case with language, context is a decisive factor in how a term will be interpreted; and context provides crucial clues to the author's intentions.
And finally, the only advice I can offer on the question of when you might use it appropriately is to play it by ear. Look at instances where others have used the term, consider how their usage was received by others, and then decide whether you are comfortable using it in similar situations. In general, I think, you are better off using an expression whose nuances you thoroughly understand than one that you have only a shaky grasp of.