A previous answer to "how" offered the Second World War, but that's facile and incorrect. The answer lies in President Harry Truman's second term. Take a look at this simple chart of military spending to see what I mean. You can also look at this more detailed explanation (PDF) of outlays and military manpower by the U.S. Census between 1945 and 1973. What you'll see is an enormous dip after 1945. In absolute terms, it was the largest demobilization of military manpower in American history (proportionally, the American Civil War demobilization was larger). Why, then, did it increase?

There are three principal reasons: One is political, the second is military, and the third is technological. Let's begin with the political.

The 1948 U.S. Presidential race pitted Harry Truman, Democratic successor to Franklin Roosevelt, against Thomas Dewey, a Republican from New York. Today, Dewey is known mostly for the appearance of his name on the front page of a newspaper held by Harry Truman. In 1948, however, Dewey was a serious figure with serious aspirations to become president.

Dewey was an immensely popular governor, and he seemed to be a shoe-in for president. Truman was poorly thought of in large portions of the country ─ so much so that Strom Thurmond would split the Democratic ticket by running as a "Dixiecrat" in the South. With his victory seemingly assured, Dewey didn't run a very enthusiastic campaign. He came across as stilted and inoffensive, bland and vanilla. Truman, on the other hand, was an enthusiastic and energetic campaigner, something that would prove to be the deciding factor in the race.

The pair were also divided ideologically. Truman was the inheritor of the Rooseveltian internationalist platform. He believed that the United States had a duty to intervene around the world where Democracy was threatened. He supported the plan put forward by Secretary of State George C. Marshall to spend billions of dollars on aid to Europe. The Marshall Plan, as it came to be known, was passed by Congress and signed by Truman in the first half of 1948.

Similarly, Truman supported anticommunist forces in the Greek and Chinese civil wars, and he put diplomatic pressure behind anticolonial movements around the world. Conversely, the Republican platform in 1948 favored less intervention overseas and greater focus on domestic issues. While Dewey, from New York, was definitely on the liberal wing of the Republican Party of the time (he supported expanded Social Security and the first steps toward national health care), he had to keep the support of men such as Sen. Robert Taft of Ohio, the leader of the conservative Republicans. Dropped into today's political scene, Taft would be considered more a libertarian ─ he favored rolling back many New Deal programs, cutting taxes, and cutting the military. In 1951, Taft's book, A Foreign Policy for Americans stated:

"Force should not be called for against any nation because of any internal domestic policy, except rearmament in excess of a quota imposed or agreed to. Interference in domestic policies, even such vital matters as tariffs or the treatment of minorities, would be more likely to make war than to prevent it. The test is: is the subject one on which the people of the United States would be willing to have other nations interfere with our internal actions? If not, we should not attempt to impose such interference on others."

What Dewey would have done as president internationally is a what-if question we'll never know the answer to. Nevertheless, we can say with certainty that Truman, who believed in international intervention, was much more likely to increase military spending than Dewey was.

Militarily, Truman's second term was an eventful one, and this may be the key factor in American military expansion. On Aug. 29, 1949, the Soviet Union tested its first atomic explosive. Ten months later, North Korea ─ with the support of the Soviet Union ─ invaded South Korea to begin the Korean War. In both instances, the United States was grossly unprepared for events.

/u/restricteddata could explain more, but contemporary experts thought the Soviet Union was years away from detonating its first atomic weapon. When it happened in 1949, it was abrupt, sudden, and terrifying from the American point of view. Between 1945 and 1949, the defense of the United States had rested on the principle that the American atomic monopoly could and would balance out any disparity in conventional arms. The atomic bomb was the great equalizer, it was thought, and so the United States did not need large numbers of conventional arms or enlisted soldiers to fight on a traditional battlefield. The twin detonations of Hiroshima and Nagasaki, coming shortly before the Japanese surrender, in hindsight created a myth about the atomic bomb: It was so terrible that any enemy could not continue fighting against it.

When the Soviet Union detonated its weapon (and that weapon's existence was exposed by air-sampling aircraft from Alaska), that theorized equalizer was gone.

When the Korean War began, it seemed to prove to Americans that not only was the equalizer gone, but the Soviet Union was willing to exploit the situation. American soldiers, rushed to Korea to stop the invasion, were outnumbered, outgunned and totally overwhelmed. In one example, American soldiers found their 2.75" WWII-era bazookas unable to penetrate the front glacis of North Korean tanks. America rushed to rearm and redeploy, even going so far as to remove tanks from static display in front of National Guard armories and send them to Korea.

Edit: For more on the military aspect, read /u/GTFErinyes' posts below, and in particular his emphasis on Marshall's ideas that future conflicts would not allow a long period of mobilization.

This leads into the third issue: Technology. If you look at those Census charts way back at the start of this comment, you'll see that although there's a surge in military manpower after 1950, it never reaches WWII levels. Spending, however, skyrockets. Since the end of World War II, the United States has embraced a philosophy of technological superiority rather than numerical superiority. Rather than deploy large numbers of soldiers to the battlefield, the U.S. military has chosen to deploy fewer numbers of well-trained and well-equipped fighters.

This has a cost, of course. Newer material always costs more, and research and development costs are highest at the leading edge of technology. With the world's largest economy, the United States is able to support this military spending, and even though it is large in absolute figures, it has tended to be relatively small in terms of gross domestic product. The U.S. economy is so large that even though the U.S. is far and away the largest military spender in absolute terms, its 3.5 percent share of GDP is today tied for the 20th largest in the world.

When the United States originally (shortly after the Declaration of Independence) was created, it was based on the Articles of Confederation. It was more an alliance of the different states/colonies, rather than a Union. There was not really a central government, no commander of chief, and no standing army. Basically, the responsibility of having military was within each state and it basically said "have an army, but not too large and don't attack your neighbors."

but every State shall always keep up a well-regulated and disciplined militia, sufficiently armed and accoutered, and shall provide and constantly have ready for use, in public stores, a due number of filed pieces and tents, and a proper quantity of arms, ammunition and camp equipage.

http://avalon.law.yale.edu/18th_century/artconf.asp

In plain English, the Articles of Confederation sucked for several reasons that I won't go into now. It was later replaced by the United States Constitution (instead of trying to fix the AoC). You need to understand, that when the AoC were written, it was just at the time when they got independent from a monarchy. They were very anxious about having a central government with a central army "on the top," let alone a person such as the commander in chief.

Then later, the US constitution states:

The Congress shall have Power ....

To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water;

To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;

To provide and maintain a Navy;

To make Rules for the Government and Regulation of the land and naval Forces; To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions;

Note that the President has power over the militia of the states:

The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States

As you can see the power over the military was moved from within the states to this new central government. However, the states still have indirect power on it through the congress.

https://www.constituteproject.org/constitution/United_States_of_America_1992

I am not sure if this answers your question 100%, but having a military always seemed to be a significant part of the United States nationality. The "city upon the hill" needs to be defended and many times people were trying to expand the idea of the republican government (not to be confused with today's Republicans) to all over the world. I am sure that the civil war, Mexican War, etc. also added importance to having a standing army that can defend the Union.

Samuel Huntington's The Soldier and the State is one of the preeminent works on civil-military relations. In it, he discusses the ideas of subjective and objective civilian control over of the military.

• Subjective civilian control is maximizing civilian power in relation to the military; when one group of civilians is empowered at the expense of another group of civilians.

• Objective civilian control is maximizing military professionalism; making the military a tool of the state rather than a mirror of the state; this makes them politically neutral and sterile.

How did the Framers view the question of civilian control of the military? Why does the constitution make civilian control of the military difficult to achieve?

• The Constitution does not permit the objective civilian control compatible with a high level of military professionalism

• The Framers wrote a subjective approach to civil-military relations; military professionalism and objective civilian control was unknown in the 18th century

• The Framers' concept was to control the uses to which civilians might put military force rather than to control the military themselves

If the Constitution was written 30 years later, it is very possible that the language in it about the military would be written with objective military control in mind. Additionally, the Framers were interested in subjective control / "suspicious of standing armies" because of their experience with how England used its military as other posters in this thread have explained.

From Huntington's book (pg. 189-190):

Objective civilian control has existed in the United States but it has been the product of geographical isolation and the international balance of power, which permitted the virtual elimination of standing military forces and the exclusion of the military from political power. Civilian control in this sense has been so effective that Americans have called it a fundamental principle of their system of government. But they have been deluding themselves. They have ascribed to the Constitution a virtue of geography. Objective control has been extraconstitutional, a part of our political tradition but not of our constitutional tradition. Civilian control has, in a sense, been like the party system. The Framers did not foresee the rise of popular democracy; consequently, they did not provide for political parties. They did not foresee the rise of the military profession; consequently, they did not provide for civilian control.

Follow-up: how has the US managed to keep military influence in politics to basically zero? I live in Thailand and the military has the same monopoly on force but does not answer to any civilians. The result is they take over the country periodically. It's a pretty common situation in developing countries and I'm curious how the US avoided that.

There is a good case to be made that the the turning point was the Korean War.

While the World Wars marked huge mobilization efforts and military spending, the United States has up to that point demobilized and Military defense spending extremely quickly.

The Civil War : In 1864, military spending was 781 million dollars, 5 years later, military spending was down to just 127 million dollars.

WW1 : In FY 1918, military spending was 12,007 million dollars. Just 4 years later, FY 1922 military spending was 1,303 million dollars.

WW2 : FY 1944 and 1945 had defense spending of 79,143 million and 82,965 million respectively, yet two years later, FY 1948 allocated just 9,105 million dollars for defense, a 89% decrease.

Despite it's magnitude, WW2 does not seem to have broken the "rule" or rapid American demilitarization after a major conflict. President Truman made cutting defense spending one of his primary goals, and rapidly demobilized the armed forces. One of the key "benefits" of focusing on the nuclear force was the belief that it made conventional forces obsolete, and thus could save money by replacing large amounts of conventional capability with a smaller and cheaper nuclear force.

More spending was allocated in 1949 due to the fall of Nationalist China, the Berlin Airlift, and the Soviet Union detonating it's own Nuclear Weapon, which put some doubt in the absolute power of the USA's atomic arsenal.

The Korean War was a shock to Truman government. Without funding, the conventional capability of the US military had atrophied. When Truman called for a blockade of North Korea, he was informed that the US Navy could only implement a "paper blockade"; the victorious US Navy of 833 warships in 1945 had only 161 on hand in 1950. The nuclear arsenal could not be used without severe consequences, and the US would have to rebuild it's conventional forces to fight the Korean War.

The Korean War was the point when the US realized that the Cold War can turn "hot", and the limits of Nuclear weapons. It would need a large standing army in order to have the capability to counter the Soviets both in Europe and elsewhere around the world.

The need for further US involvement was cemented by the French defeat at Dien Bien Phu in 1954. The colonial powers of Europe were spent by the World Wars, and would be unable to police their soon-to-be former colonies.

Edit : see /u/The_Alaskan 's post [here] (https://www.reddit.com/r/AskHistorians/comments/3wlmpa/how_and_when_did_the_us_go_from_a_nation_with_a/cxxaxzw) for more details on the political side.

Sources

US navy active force http://www.history.navy.mil/research/histories/ship-histories/us-ship-force-levels.html

US historical budgets https://www.gpo.gov/fdsys/pkg/BUDGET-2016-TAB/pdf/BUDGET-2016-TAB.pdf

"Blockade on Paper" http://www.trumanlibrary.org/whistlestop/study_collections/korea/large/week2/kw_78_1.jpg