Well first of all you really need to define whether we are each other's strongest allies. America has incredibly important strategic relationships with Israel, Japan and Saudi Arabia and also Canada, which is potentially America's strongest ally but flies under the radar somewhat. Looking at it this I'd argue that we're probably #2 after Canada, mainly because of cultural and historical reasons and our military's ability to project power (unless the current government has it's way). Saudi Arabia is an alliance of conveience, Israel is important from a political-military standpoint but not culturally unless you're Jewish American while Japan is hobbled by it's constitution - although it will become increasingly important as China rises. While the UK is far less strategically important than it was during the Cold War era it's still a very important partner for the States.

Anyway having established that how did we get to where we are today? Possibly the first building block of this relationship in my opinion goes right back to the treaty that ended the Revolutionary War. In the Treaty of Paris (1783) Britain ceded the entirety of it's Trans-Appalachian land holdings to the new American republic. Britain did not have to do this, control over the region was highly contested and Westminster had never recognised the various colonies' claims to this land previous to the war. The French government had expected the British to attempt to keep control of this land thus leading to future wars but the British wisely ceded it with of the aim of preventing future hostilities and keeping America within their economic sphere of influence, reasoning correctly that trade with the colonies would remain lucrative after the war ended. This generous act not only pleased the Americans but removed a huge potential conflict point between the two nations that could have rumbled on for decades poisoning relations.

While the Treaty of Paris was generous in it's terms and a good starting point for repairing relations, the British Army was rather tardy in removing it's forces from this land and they would remain there until the Jay Treaty (1794) where the Federalists acting largely on the initiative of Washington and Hamilton gained Britain's acquiescence to remove their forces. In return Britain was granted 'most favoured nation' status, the United States co-operated in anti French naval actions and also paid off some pre-Revolutionary debts to British merchants. Once again it proved the benefits of co-operation instead of conflict but coming only eleven years after the end of the war it was highly unpopular and there were many protests against the Treaty throughout the country. HBO's TV series John Adams actually covers this quite well from my recollection.

After Jay, relations were rather close from a commercial perspective if never warm politically as the White House had occupants rather less pro-British than Washington/Hamilton/Adams. Jefferson's Presidency was marked by his anti British beliefs as was his successor Madison, under whom the two nations slipped back into war. During this war the Americans would march north and burn York (Toronto) and the British retaliated by landing on the Potomac, swatting aside a hastily assembled American army and marching into Washington D.C. and setting it alight. The war didn't really achieve anything and led largely to a status quo ante bellum however it was the last time the two countries went to war.

After 1812-15 relations for the rest of the century went back to the style of the Jay Treaty, extensive commercial relations continued developing and whenever issues flared up they were resolved via mutual discussion and the signing of mutually beneficial treaties. The Canadian-American border was demarcated in such a way as was the division of the Oregon Territory in 1846. Although tensions manifested themselves at times such as over the Oregon Territory when some members of the American establishment wanted a third war to take the entire territory (54' 40' or Fight!) and during the Venezuelan Crisis in the closing decade of the 19th century, cooler heads tended to prevail, as they did when Britain decided to stay out of the American Civil War. During this period the British economically dominated the United States, in the early part of the century they had almost a colonial relationship (think late 19th century Britain-Argentina) and even as the American economy began to grow and diversify the British still held a lot of control over it right until the end of the century. Extensive British migration continued and the British as a whole were still the dominant culture within the United States alongside the Germans right until the closing of the century. Although this obviously helped it should still be remembered that the Americans were virilently anti-monarchial (ironic given their present love of the British royal family) and bristled at any perceived inferiority to the former mother country.

Fast forward to the end of the century and in 1898 the United States beginning it's emergence as a true world power goes to war with the decaying Spanish Empire. The Old World powers of Europe almost universally criticise the upstart Americans except for one power that is conspicuous in it's silence...Britain. The United States then reciprocates this in being rather quiet regarding the British war in South Africa in 1899-1902. Some historians date the development of the 'Special-Relationship' as far back as this rapprochement but even if you don't want to do that it's clear the two nations were drawing closer together. As Otto von Bismarck quipped, the most important factor of the 20th century would be that 'North America speaks English'. By this point America had also grown as a nation, it no longer needed to measure itself against Britain or hang on too aggressively to the old Revolutionary slogans and beliefs. Close relations between the two were often promoted on both sides of the Atlantic, Conan Doyle had Sherlock Holmes wondering whether the two nations would perhaps merge into one again. As a result such tacit agreements and strengthening of relations would no longer result in the protests and ill tempered debate that Jay had created a century earlier. For a good starting point on this concept look here (http://en.wikipedia.org/wiki/The_Great_Rapprochement -> Obviously it's Wikipedia blah blah blah but it's a good spring board for this aspect of Anglo-American historiography).

Following this period we have of course WW1 in which the British and Americans fought side by side against Germany. Well kind of anyway...the vast bulk of the American Expeditionary Force was deployed far to the south with the French in between the two armies but some Americans did serve alongside the British in '18 especially in breaking the Hindenburg Line. However imo this era is more important for tying the two economies even closer together, interestingly many of the major British commercial interests that had been built up in the USA over the 19th century were liquidated to pay for the war and America ended up loaning Britain incredible amounts of money turning America from a major debtor of the United Kingdom into a creditor of her. Quite the opposite of the previous century's relationship.

After the war America returns back to it's isolationist nature as President Wilson mishandles it's supposed entry into the League of Nations. We have a number of largely uninteresting years marked only by a visit to the States of members of the Royal Family in the 1930s and the signing of the Washington Treaty to prevent a naval arms race, which is entirely within the spirit of Anglo-American reciprocal treaties of the 19th century. Then comes WW2 which I'm not going to get into too much as everyone knows that story off by heart and all about the American 'occupation' of Britain, giving British women chocolate rations, D-Day and so on.

World War Two ends with Britain half destroyed and exhausted but with the two nations sharing a common enemy in the revisionist power of the Soviet Union. Throughout the late 40s and early 50s Britain attempts to maintain some vestiges of it's previous power but ultimately slowly settles into the role of America's lieutenant, it's strategic outlook, geographic position and shared political environment makes it ideally suited to this role and its a role that Britain has occupied pretty much ever since. The fact that Britain recognised the situation on the ground, much as it had in the 19th century and acquiesced to this change in status helped speed the process along, although road bumps were encountered along the way such as Britain's refusal to send troops to Vietnam. Had Britain more virulently resisted it's decline in power or attempted to steer a third way such as De Gaulle's France then the relationship may have developed differently. Even British entry into the EC (later EU) didn't change their relationship, this being supported by the United States and Britain being a very useful source of American influence within the EC as De Gaulle had actually predicted when twice vetoing Britain joining.

At the same time the two cultures which had diverged somewhat during the 19th century began to converge once again. A shared language and the emergence of transatlantic media allowed the cross-pollination of much of American and British media, Elvis Presley was massive in Britain, in the 1960s you had the 'British invasion' of America spearheaded by the Beatles and so on. With the advent of the internet this phenomenon has only increased in it's depth and breath. British shows such as the Inbetweeners and Downton Abbey are huge in America, British artists such as Ed Sheeran regularly top the American charts and British newspapers The Guardian and The Daily Mail are amongst the most read newspapers online in the United States. Likewise American movies are still as popular as ever in the UK, Beyonce and Rihanna dominate our radios and one of the best selling series of fictional books in the UK A Song of Ice and Fire is written by an American author. The internet has expanded this cross-pollination massively by allowing Brits and Americans to extremely easily exchange ideas and content and also in things like chatrooms, Reddit and Skype easily socialise (it should also be mentioned that this applies to the rest of the Anglosphere too). Ironically this tightening of culture and media has occurred at the same time as revised American immigration laws have seen the face of the United States change immensely, as the American population becomes more Latin American and Asian influenced and less Anglo-German influenced in terms of it's ethnicity, while Britain has absorbed a large influx of Asian, African and Eastern European immigrants.

tl;dr This subject is incredibly detailed and spans hundreds of years of history and disciplines and I really quickly ran through some of the salient points. If I had to explain exactly why the United States and the United Kingdom have become so close it'd be a number of reasons. The resolution of serious arguments centuries ago and long before they could fester into any serious resentment, the development of and shared commitment to mutual discussion and agreement on any disputes, a broad commitment to free trading capitalism although this has been challenged from time to time, the emergence of shared enemies (France at times, Germany, Japan, the Soviet Union, Iraq, Al-Qaeda etc) and the factor that I believe is growing more important with every passing day - the shared language and huge cross-pollination of media between the two nations.

Sources:

• Old World, New World: The Story of Britain and America by Kathleen Burk (Abacus) (2009)

• Unlikely Allies: America, Britain and the Victorian Beginnings of the Special Relationship by Duncan Andrew Campbell (Hambledon Continuum) (2007)

• 1914-1918: The History of the First World War by David Stevenson (2004) - Primarily if you want to read up on how much WW1 changed the economic relationship between the two