Q:

How did 'stump up' compound to signify 'pay up money'? 

How did 'stump' compound with 'up' in stump something up, to signify:

pay a sum of money[?] 
‘a buyer would have to stump up at least £8.5 million for the site’

Etymonline for 'stump' didn't help either.  

From 1590s as "reduce to a stump."  Sense of "walk stiffly and clumsily" is first recorded c. 1600.  Sense of "baffle, bring to a halt by obstacles or impediments" is first recorded 1807, American English, perhaps in reference to plowing newly cleared land, but compare earlier sense "to challenge, dare" (1766). 

A:

In the olden days, markets were set up in cleared areas where trees had been cut down and a tree stump could be used as a rudimentary seat or table. 

This phrase derives from horse trading or other dealing where money was put onto a wooden post (stump) to display good faith in a cash deal.  It is a variant of the expression 'cash on the nail'. 
The Reverend William Carr published a glossary of the colloquial language of the Craven district of West Yorkshire in 1828, titled The Dialect of Craven, in which he defined 'stump' as a verb:

"Stump, to pay ready money,... to pay down on the nail." 

From The Phrase Finder

A:

To stump, without the perfective up, first appears with the meaning ‘pay ready money’, i.e. cash in hand, in a Yorkshire dialect dictionary, The Dialect of Craven, by the Rev. William Carr, London, 1828.  The origin lies in outdoor trading for horses, cattle, etc.  where a buyer would put cash on a stump or post to show sufficient funds for a purchase, as Carr explains, ‘cash on the nail’.   Americans would more readily understand cash on the barrel arising from a similar context.  
The phrasal verb with up first appears in magazines and newspapers a few years thereafter:

“Why, true, true, Doctor---it might be as well for us all; but who's to stump up for it?”  replied one of the officers.  I gave him five shillings, and the servant was instantly dispatched for a hackney-coach.—  Robert Walsh, John Jay Smith, The Museum of Foreign Literature, Science, and Art, 1831.  Also, citing Blackwood’s Magazine, The Sydney Gazette, 11 Aug. 1831. 
… but that would not do as he had not got the requisite toggery, and no pals to stump up the blunt to rig him out.   — Westmoreland Gazette, 3 March 1832.  BNA
James Fellowes, a natty little fellow, in plush jerkin and leather unmentionables, and a bang up upper benjamin, with mother o'pearl buttons as large as saucers, had to stump up the ready for keeping it up the previous evening. —  Sydney Herald, 26 March 1832. 
... on two occasions Mr. Fyshe Palmer gave me no remuneration ; but on this occasion I expect that Mr. Walter will stump up handsomely. —  Morning Post (London), 3 Nov. 1832.  BNA
The people are well enough pleased with patriots so long as they keep their hands out of their breeches pockets; but when stump up is the cry, people double-lock their strong boxes ... — The Atlas (London), 8 Dec. 1833.  BNA

The “natty little fellow” in Sydney paying a cash fine for public inebriation in a delightful, slang-rich police report suggests an earlier origin for the expression than Craven’s dictionary might indicate. 
The first attestation of stump up in the American press hinges on no one knowing what it meant.  The context is a bare-knuckle prize fight between two rivals since childhood, both sons of Irish immigrants from the same village in Ireland: John C. Heenan, the “Bernicia Kid,” and John Morrissey.  Since bare-knuckle fights were illegal in the US, the bout was held across the border in Canada, 20 Oct. 1858.  Heenen’s left hand was injured on a post, so that he lost the bout, which means that a considerable number of bets were also lost.  The humorous article shows people of various social classes — a domestic servant, two boys who bet marbles, and two physicians — discussing this very thing:

Doctor Q — Well, Doc — fork---hand over-“stump up.” 
  Doctor Z — Stump fork !  what do you mean ? 
  Z — Benicia’s busted up — that’s what I mean.  Hand over that order.   — “The Late Prize Fight,” Rockland County Journal (Nyack NY),  4 Dec. 1858.