A large number of things happened over several hundred years, which is quite a long time. No simple answer here, my friend, but I will point out several major advantages that the United States has over others.

The first one, one that isn't particularly obvious, is the fact that it is isolated. The nation, since its origins, has been located on what is essentially the far side of this planet, over a particularly stormy ocean. For most of its history, it has been difficult to reach by other nations, whether by sea, or eventually, by air.

This affords a precious thing: security. It is hard for another nation to invade and wage war against America, as it doing so would require the deployment of a sizable navy to ferry soldiers and supplies over, not to mention the logistical and command nightmares that such a long distance would cause. As such, a European nation might have to spend considerable time and effort fending off constant warfare, America has the safety of isolation.

The second point, which shares quite a bit of similarities with the first, is that America has room to grow in due to being located on a continent without other major powers. I'm not saying it is 'unpopulated', since that would be discounting the native peoples living there, but keep in mind that the natives had suffered significant population collapse in the previous century or so, and had difficulty opposing the settlers of the eastern coast.

As such, the American population could grow at an alarming rate. As immigrants and natural child-birth drives the population up, more land is simply claimed and settled, and more natural resources are tapped to sustain the growth.

Contrast with Europe, where significant settlement and resource exploitation had already taken place, and excess population simply increased the density of existing areas of civilization, and there are no 'unclaimed wilderness' to expand into. The only thing that could be done was to farm more intensively to get more food out of the land, rather than getting more land. In short, Europe was 'full'.

With those two points in mind, it should be sort of believable that the United States boasted a huge population, powerful economy, and significant reserves of natural resources in only a few hundred years. None of it was due to some particular virtue of the American people (no offense), but rather due to excellent geography.

If you wish, you could even compare the situation with Russia. Russia had similar degrees of isolation rendering some semblance of security, and a vast wilderness full of resources to expand into. These advantages allowed for the growth of the USSR into another superpower to rival the United States.

Of course, geography is not the only factor. There were a few more things, but chances are, 'pulling yourselves up by your bootstraps' and 'good ol' American ingenuity' or 'DEMOCRACY!' played a smaller role than you might believe.

There are several key factors to the US' growth, including the isolation and growing room that /u/SlavicThunder pointed out, but let me elaborate on that.

The United States has the largest amount of useable land in the world. That, and the great distance from Europe provided Americans with the opportunity to take advantage of the land, and expand west. "Manifest Destiny" ideas were strong in the 1800s and helped escalate the expansion of the United States. And to further strengthen the isolation between the US and the European world, the Monroe Doctrine was created, basically telling European powers to stay out of North and South America. This idea is still used today whenever the US intervenes in Latin America. At the time of the Doctrine's signing, though, the US did not have the power to enforce it; instead, the British navy helped quite a bit.

Examining US history, you'll find that the Brits and Americans have had a strong bond since the war of 1812, which has benefited both nations in many ways. With the industrial revolution beginning in Britain, it immediately hit the United States, and at the perfect time. In the 19th century, American coal, steel, paper, cottonseed oil, and copper wire production all increased at least tenfold. In 1894, the United States displaced Britain as the world's leading manufacturer. "By 1914, it was the world's leading producer of coal, natural gas, oil, copper, iron ore, and silver, and its factories were producing more goods than those of Britain, Germany, and France together."

In 1850, America's 23 million people had a cumulative wealth of $7.1 billion. Fifty years later, the population had tripled to 76 million, but the wealth had increased thirteenfold to $94.3 billion. This growth was not only due to the goods produced, but the patents for the goods. In the 1800s, the US Patent Office approved some of the most useful and profitable patents in the world. The passenger elevator, escalator, telephone, phonograph, air brake, cash register, electric light, fountain pen, box camera, adding machine, revolving door, safety pin, paper clip, and the typewriter were all invented in America in the 19th century.

All this could not have happened without immigrants. Between 1815 and 1915, the US took in 35 million people. The mixing of different peoples allows for the mixing of ideas, and the tolerance of different cultures and religions, which help a nation grow. The log cabin, an image of American pioneers, actually came from Swedish immigrants. Hollywood was largely created by Jewish immigrants.

Lastly, and most importantly, the World Wars had enormous impacts on the United States' growth. In both wars, the United States got involved late, and never faced destruction on its own land--except Pearl Harbor. Thus, at the end of both wars, the United States stood on top while the rest of the world struggled to pick up the pieces of their country. To secure their position as the world power, the US did two major things:

First, the US was made a permanent member of the United Nations Security Council.

Second, the Marshall Plan or The European Recovery Program. At the end of the second World War, the United States gave over $13 billion in loans to help reconstruct Western Europe, with one catch: The United States would be allowed to establish military bases all across Europe. This, in addition to the bases that the US has all over the world, contributes to the great military power.

Source: Made in America by Bill Bryson and University classes.