To understand the fear that lead to 'Domino Theory' one needs too understand the idealogical underpinnings of Marxism. In the Communist Manifesto, Marx predicted that the Workers Revolution would be a worldwide affair. The U.S.A acknowledged this to a certain extent by Domino Theory, seeing it as the potential start of a truly global revolution, spreading from former Russian Empire, to China, and then throughout Indochina and eventually onwards across the globe.
Also the potential for an Asiatic bloc of Communist powers was understandably terrifying to the USA due to the nuclear arms race of the 50s and 60s.

Another thing to consider is that the Cold War was a war of ideology. People genuinely believed that their method was the only way to live, either through Western ideals of liberal democracy, or those espoused by Marx and Lenin.

The U.S. cared because they had been employing the policy of "containment," the idea that global communism must be subdued. The Soviet Union, under the ideology of Stalinism, wanted a worldwide communist revolution because only then would there be a proletariat state without class struggle and maintained by class equality and socialism. This ideology conflicted with the Americans. The U.S. believed that capitalism, but not necessarily pure capitalism, was the most functional model of economics. Communism is a direct rejection of capitalism. Communists believed that in order to constructed a pure communist world, the proletariat must rise against capitalists, destroy capitalism, and develop a purely proletariat state. Ideologically, this is a threat to the U.S.. American-Soviet ambassador George Kennan warned President Truman about the Soviet's masterplan. Kennan thought the Soviet ideology was a threat to traditional American ideals and a poor representation of the Russian people's beliefs. Moreover, he believed that the Soviet government was oppressive regime with a delusional philosophy. Truman took Kennan's advise and established his "Truman Doctrine" which warned the American people about the Soviet's totalitarian ideology.

National ideology, however, was not the only reason the U.S. joined in the Vietnam War. It may be the main reason, but the U.S. also faced a lot of domestic political pressure in its decision to fight, and remain in, a war against the Vietcong.

Even if they didn't really care so much for those countries, after they fall what next? Their ally Australia? 1 billion people in India? Japan? As dominoes continue to fall, they come closer and closer to places that might have even more strategic importance.