This is the subject of an important book by John Brewer called The Sinews of Power. He tracks the ways that the British state was able to more or less win the series of wars against France from 1689 to 1815 because it developed the machinery necessary to finance them. In that period, the state's debt essentially doubled with every war, not just the American war that you asked about. The banking establishment of the City of London and the British state collaborated to develop what Brewer calls a "fiscal-military state," a bureaucratic and governmental apparatus designed to collect taxes, borrow money, and fight wars. Particularly important in this area and largely distinct from France were the growth of a professional bureaucracy and what were called funded loans. The state borrowed money from London financiers, but guaranteed that loan by tying it to the collection of a new excise tax. Thus, a loan needed to fund new ships would be tied to a tax on the production of, say, candles, collected by an increasingly professional bureaucracy over a period of time. This practice allowed them to turn short-term debt into long-term debt, essentially creating a running national debt; they gave up trying to ever pay off the whole thing by about the 1720s, and concentrated just on paying the interest.
One interesting thing about this fiscal-military state is that it actually helped to tie the United Kingdom together. England and Scotland had only been united with the Act of Union in 1707 and the issue was far from settled. The fiscal-military state provided an avenue for investment in the state, but that very investment gave many people of wealth in both kingdoms very clear interests in maintaining the new United Kingdom. Allowing England and Scotland to fracture into war again--as might have happened if, say, the 1745 Jacobite uprising were successful--would have jeopardized their investments. So, the long series of wars against France provided means for the integration of institutions, creating a substantial and powerful class of people with deep interests in keeping the Union successful.
To add to what /u/agentdcf says about Britain's advanced fiscal-military state, there were a few intrinsic advantages Britain had which helped her financial situation--the biggest of which was her reliance on the Royal Navy, rather than the army, to be the main arm of the military and defend her territory. Britain, as an island nation, did not have to fear a major land invasion and there were several strategic developments which cemented this: the Act of Union in 1707, the final defeat of the Jacobites in 1745 and the 1756 alliance of France with Austria, which protected the Netherlands, and their ports, from domination by either power and absolved Britain from her only possible continental military commitment. From that point on, Britain could maintain the bare minimum of an army and guarantee her security with the navy alone. Between 1783 and 1793 the army 'sank into dereliction',^1 but this was acceptable because there were no serious invasion attempts. Britain 'needed only enough trained soldiers to force an opponent contemplating invasion to mount a large effort for which the preparations could not fail to be detected early enough for the Royal Navy to intercept it'.^2 At the same time, every other European power had land borders to defend.
The crucial factor here was that the navy was much better value than the army. Although it was extremely expensive to build ships, but only around thirty had to be kept ready for emergencies. In peacetime the rest of the navy was mothballed, dockyard manpower establishments reduced by two-thirds and the sailors released to maintain their skills working on merchant ships. Armies could not be scaled in this way--if troops were disbanded, it could take months or years to raise new units of a similar calibre, since there were few civilian trades which required the skills of soldiering. Continental armies had to be always maintained at a useful size, and while in wartime the cost was reduced because armies could partly support themselves off enemy territory, this was not possible in peacetime, and the cost was enormous. Britain also had a large base of skilled seamen--116,000 in 1783, compared to 63,000 in France in 1789--and good shipbuilding facilities due to thriving maritime commerce, which was encouraged by the economic policies of successive governments (the Navigation Acts particularly) and various naval efforts to facilitate trade (the Anglo-Dutch wars and expeditions to Denmark). Most taxation was indirect, due to the popular mistrust of direct taxation (which had been one of the causes of the Civil War), and mainly took the form of customs and excise. Thus the navy, the primary source of military strength, and maritime commerce, which underpinned British finances, were mutually supportive. No such clear symbiotic relationship existed where land armies were concerned; nor even in the continental navies, because they had less commerce and because their warships tended to be larger and thus harder for private shipbuilding facilities to handle. Because Britain could be dependent on her navy, it could develop more quickly--'no other European state possessed this utter conviction of the importance of maritime power to its very existence'--^3 so Britain was the first nation to create a code of naval discipline (1653), naval academy (1729) and adopt victualling ships and copper bottoming technology, which put her even further ahead of her rivals. And the British were able to foster, though a broadly meritocratic system, a famously aggressive, professional and innovative spirit in the Royal Navy.
So this is a long-winded way of saying: Britain relied upon the navy, the navy was cheap, and the better the navy was the more money there was to fund the navy. This doesn't mean Britain's finances were necessarily rosy--it was a consistent aim of peacetime governments to reduce the national debt--but it gave her an edge over the competition. Even Britain probably wouldn't have been able to maintain both a large army and large navy--the navy was starved of funds during the American War of Independence to cover the cost of the land war--but she didn't have to. When others tried--especially France after 1763--to maintain their armies and expand their navies, they simply ran out of money (which was a contributing factor to the French Revolution). Britain had by the late eighteenth century an advanced, centralised fiscal state bureaucracy which usually wasn't actually taxed as hard as it could have been because Britain could focus on her navy and didn't need to support a large army.
Sources
Also: Jeremy Black, Naval Power: A History of Warfare and the Sea from 1500 (Basingstoke, 2009). John Brewer, The Sinews of Power: War, Money and the English State 1688-1783 (London, 1989). Christopher Storrs (ed.), The Fiscal-Military State in Eighteenth-Century Europe, Farnham, 2009).
n.b. I've replied to a comment elsewhere about the situation in other countries.