The short answer is ‘we won the war(s).’
The longer answer is that after the American occupations expired in Germany and Japan after WWII we were in the middle of the cold war and neither Japan nor western Germany were in any mood to become Soviet citizens. So, the US entered into something called a ‘status of forces agreement’ with the local countries to retain a US military presence. It is why we still have bases in England long after WWII.
The long term strategic goal is that the NATO countries plus Australia can respond to two major military engagements anywhere in the world. We are really tied up in this, more than is popularly known by the common individual. NATO militaries would work much less effectively without an American presence. The USA trains and supplies most NATO militaries in some way or another. They do exercises and deploy together. When you start respecting that NATO + Australia is really one large military it becomes obvious why you need American bases everywhere.
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