Why do the British refer to things as '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:

Port out
Starboard home
Posh with a capital P-O-S-H, Posh!