Why do the British refer to something very smart, or people who are very well-off as being 'posh'?
There are plenty of apocryphal stories. Given that the word does not have any lingering connotations of travel or tickets or navigation the acronym hypothesis is far-fetched and very unlikely.
The most likely origin of this word is from London street slang for money, current in the nineteenth century. It is a Romani word posh for halfpenny or small amount of cash (posh-kooroona for half-crown and posh-houri for halfpenny). This is likely the basis of the slang use of this word for cocaine to indicate the pricey nature of this commodity.
One of the earliest references (1892) that conveys the sense of dandy and swell again originates in London and came from George and Grossmith's character in The Diary of a Nobody.
There is a detailed analysis available here
Phrasefinder provides the following interesting information:
The first recording of 'posh' in print that seems unequivocally to fit the current meaning of the word is a cartoon which contains this dialogue between an RAF officer and his mother, also in Punch, September 1918:
Oh, yes, Mater, we had a posh time of it down there."
"Whatever do you mean by 'posh', Gerald?"
"Don't you know? It's slang for 'swish'"
The OED suggests that it might be related either to the noun posh, now rare, meaning money or to another noun posh also slang, and equally rare, meaning a dandy. The explanation continues:
. . . the semantic development may thus have been either from ‘money’ to ‘moneyed, wealthy’, and hence to ‘upper-class’ and ‘smart, stylish, luxurious’, or alternatively from ‘dandy’ to ‘upper-class‘ and ‘smart, stylish, luxurious’.
Of the popular notion that the Peninsular & Oriental Steam Navigation Company stamped tickets for favoured cabins on the route to and from India with the letters P.O.S.H., the OED says that ‘no evidence has been found for the existence of such tickets.’
The usual story is that it stands for Port Out, Starboard Home. Passengers on big ocean liners would change cabins from one side of the ship to the other to avoid the hot sun on outward and return journeys - but only those rich enough could afford to do this.
Lionel Jeffries sings a song with those lyrics in the 1968 film Chitty Chitty Bang Bang:
Posh with a capital P-O-S-H, Posh!
Albert Barrere & Charles Leland, A Dictionary of Slang, Jargon & Cant, volume 2 (1890) has this entry for posh:
Posh (society) modern term for money, originally used for a halfpenny or small coin. From the gypsy pash or posh, a half. In Romany poshero, the affix ero being corrupted from hāro, copper, i.e., a copper or a penny. Posh an' posh, half and half, applied to those who are of mixed blood, or half gypsey. Also a dandy.
A very similar entry appears in J.S. Farmer & W.E. Henley, Slang & Its Analogues, volume 5 (1902):
Posh, subs. (thieves).—1. Money. : generic, but specifically, a half-penny or other small coin : see RHINO [for other terms referring to money]. [Citations from 1888, 1891, and 1893 omitted.] 2. (society).—A dandy.
Somewhat earlier, James Halliwell, A Dictionary of Archaic & Provincial Words, Obsolete Phrases, Proverbs, and Ancient Customs, volume 2 (1852) has this brief entry for posh:
POSH. A great quantity. West.
And John Hotten, A Dictionary of Modern Slang, Cant, And, Vulgar Words, second edition (1864) has this entry:
POSH, a halfpenny, or trifling coin. Also a generic term for money.
The 1859 edition of Hotten has simply "POSH, a half-penny, or trifling coin."
Other nineteenth-century senses of posh, such as the Northern and Midlands English dialect meaning of "a sodden meadow," are fairly clearly unrelated to the current meaning. I also could not find any early instances of the phrase "port out starboard home, which has been cited frequently as the source of posh as an acronym. Indeed the earliest match for "port out starboard home that a Google Books search finds is a book by Anna Sproule, Port Out, Starboard Home: The Rise and Fall of the Ocean Passage (1978).
The only possible link to posh in the sense of "dandy" that shows up outside slang dictionaries in Google Books' nineteenth-century offerings is the far-fetched possibility of a connection suggested by these entries from George Temple, A Glossary of Indian Terms Relating to Religion, Customs, Government, Land, and Other Terms and Words in Common Use (1897):
Posh Covering, clothed in (used in compos.).
Safaid-posh A man dressed in white, a man who can afford good clothing.
Tang-posh Tightly dressed, wearing tight clothes.
Tang-posh-ī. Tightness of dress.
In Beau Brummel's day (he died in 1840), tight clothing was a hallmark of the English dandy, but whether that was still the case in 1890, when Barrere & Leland first note the use of posh as a synonym for dandy, and whether tang-posh (or safaid-posh) traveled back to England at some point in the 1800s with returning civil servants or military men to provide a source for posh in the modern sense is purely speculative and not corroborated in any of the sources I checked. In any case, the lack of instances of posh as dandy in the wild is a major obstacle to tracing the origin of the usage.
A reasonably thorough assessment of the origin of posh appears in Michael Quinion, Port Out Starboard Home: The Fascinating Stories We Tell About the Words We Use (2005). Quinion ultimately favors the derivation through the Romany posh meaning half, applied to the halfpenny and thence to money in general.
Whether Port Out Starboard Home tickets were ever issued or not, and one supposes that if they were, hard evidence would have been discovered by now, it is a good apocryphal story and perhaps one of those things which is unimportant enough to be worth keeping as a bit of national folklore. But more importantly the word 'posh' exists because there is clearly a role for such a word in British society. It is a very British word and seems to reflect much about us, both the posh among our number and the un-posh who use the term as one of mild derision. It says more than that someone is rich or powerful. To be truly 'posh' you also need charm, wit and style as well as a snotty manner with those you consider your inferiors. Among the other 55 countries which use English as an official language, how many use 'posh', and if not how does it get translated? Has anyone ever met a 'posh American'? One hears of rich Americans, brash Americans, but I've never heard one described as 'posh'. Who in Australia might be said to be posh? In short, how does poshness translate to other parts of the Anglosphere?