I think that this is quite a pertinent and interesting question. Depending on one's location, Americans may seem to have a 'natural' predisposition towards cars and using them. However, this obviously reflects something in how that country developed and what choices it made.
In short, the re-alignment of the post-war economy with the war-time repressed demand for automobiles, combined with the post-war spending on road improvements caused the car to become ubiquitous in American society. This also trickled down to the teenagers who 'had' to have a car.
War-Time Restrictions
Firstly, what should be understood is that cars were a relatively novel thing in the early part of the 20th century. The popularization of the automobile started to progress with the advent of the affordable car as built by Henry Ford. The importance of Henry Ford and his processes—even if the Model T is perhaps overemphasised in this context—cannot be understated. Ford made it possible for the average working person to own a car; Chrysler and other companies followed. The Second World War was a small step backwards in this process:
Within two months of the Dec. 7, 1941, bombing of Pearl Harbor, the last civilian cars rolled off the assembly lines, and auto plants were frantically converting to military-only production of arms, munitions, trucks, tanks and planes. By December 1942, Detroit had become the "Arsenal of Democracy" and didn't resume civilian production of automobiles until the war ended in 1945.
—Snyder, 'No new cars, but that didn't stop U.S. automakers, dealers during WWII'
Post-War Car Demand
Similarly, car sales were restricted during the World War. However, as the Japanese surrender approached, these restrictions were loosened. This was important to both the home front people who had been wanting to buy cars throughout the war, but also the returning servicemen. At the same time, many cars sold in the immediate post-war years were the same models as on offer in 1941/42. Yet, quotas on car production were originally still in effect to conserve steel:
Up to the close of 1944, 3,451,320 passenger cars had been junked; about twice that number would have been taken off the road if they had been replaceable. The pent-up demand is variously estimated at from five to nine million new cars for the first three to five postwar years. ...
During the first three months of 1946, the industry would be permitted to produce 449,102 units. Krug said the authorized production might total 2,000,000 new cars by July 1, 1946. The purpose of the quota system was to ensure that the industry would use no more than a reasonable amount of the steel available for production of war and civilian goods. The sharp drop in military requirements after the surrender of Japan removed this factor from consideration and ceilings were lifted altogether.
—CQ Researcher, 'Automobiles in the Postwar Economy'
All this demand meant that cars were going to be bought when available. This lead to a fairly considerable car-penetration in the society throughout the post-war years but the demand was still high:
...car sales reached over 2.1 million, a figure that was slightly less than the Depression-year output of 1934. Demand was very high, but supply did not match this demand for several years even though car sales reached 5.1 million in 1949. There was a slight dip in output during the Korean War, but otherwise production continued upwards, reaching 7.9 million in 1955. By then 52.1 million cars were registered, a huge increase over the 25.8 million of the last year of the war, 1945. A decade later in 1965, car manufacturers sold 9.3 million cars and there were 75.3 million cars registered.
—Walsh, 'Gender and the Automobile in the United States'
At the same time, road network improvements were an easy way to continue mass employment while improving logistical capacity that was clearly going to be appreciated by the populace. This meant that owning a car became an imperative:
To be without a car in the United States was, except in the heart of large Eastern or Midwestern cities where a viable public transit system remained operative, to be almost in exile.
—Walsh, 'Gender and the Automobile in the United States'
Teenage Hot-Rodders & Greasers
This trickled down to the younger people as well:
Two other notable features of the mass spread of the car culture of the 1950s and 1960s were teenagers on the road and family vacations, and the overlapping dependence on roadside facilities used by both groups.
Teenagers were publicly recognized as a growing phenomenon in the 1950s and 1960s. The offspring of the parents who had married young during the war or immediately thereafter, these “baby boomers” forged their own lifestyle, a proportion of which was centred on access to the car that could give them freedom. Having wheels meant escape from prying eyes, gaining “street cred[ibility]” among high-school or college peers, and access to the local automobile culture. Gaining a driver’s licence was, as Kenneth Jackson suggests, the most important rite of passage in the high-school experience. While the majority of teenagers who inherited and customized “hand-me-down jalopies” were young males, young females were by no means debarred from teenage automobility. Indeed, they were central to this style of youth culture. Part of teenage driving was to impress other male students about knowledge of and intimacy with car technology, but another part was to be seen cruising with a popular girl, often enjoying the new rock and roll music to the annoyance of the older generation.
—Walsh, 'Gender and the Automobile in the United States'
This in turn let to the rise of hod-rodding (amongst the greasers which became a label for the sub-culture) and drag racing:
With their new found independence and parental indulgence combined many teenagers drove fancy Hot Rods. The teenager culture was enamored with the freedom a car provided. You could drive where you want, and most of the time as fast as you wanted. Some car enthusiasts would have races that became known as drag races “a preview of Saturday Night Drag Race over the loudspeakers at the Paradise Mesa Drag Strip in San Diego”.
—'Teenager Popular Culture - United States 1950s'
The following excerpt specifies the greasers as lacking opportunities but still having (tuned) cars:
Lower- and working-class families still felt the lack of economic opportunity. Minority groups such as Italian-American and Hispanic-American citizens encountered many roadblocks to prosperity during an otherwise booming period in America. Disillusioned by the lack of opportunity in post-war America, many of these people became drawn to counterculture elements like fast cars and rock and roll music. ... the greaser subculture was born. ...
Greasers were often perceived to have an extensive mechanical knowledge. With the greasers’ love of fast cars and rebellion, the hot rod was the perfect car to associate with this subculture. Racing custom-built cars became a popular pastime.
Combined with the increased prosperity and growth of suburban households, car ownership spiked. For the greaser of the 1950s, that meant owning a hot rod. A hot rod is generally considered to be a car that has been modified in an effort to produce more power and speed. ... the foundation of the National Hot Rod Association in 1951 increased the appeal for a custom-built speed machine. The mechanical knowledge of the average greaser put them at an advantage when it came to hot rods and racing their machines.
—'Greasers and Hot Rods'
Similar Sub-Cultures
The raggare in Sweden, bōsōzoku in Japan, and rockers in the United Kingdom followed some (or all) of these trends as well.
