If I recall correctly, the US maintains about 30,000 men in Germany alone and thousands more in different European countries. I understand the need to project power and to rapidly deploy to any theatre of conflict, but isn't positioning an entire army corps deep in NATO territory (now that Poland, Czech and Hungary are part of NATO/EU) a bit wasteful? I know the US can't move them to Poland without angering Russia, but why not ship everyone home and cut costs now that NATO can defend itself?
Military bases are difficult to decommission-- the host country is faced with a sudden vacuum in the local economy and a chunk of real estate that likely qualifies as a hazardous waste site, so even closing redundant bases is often opposed locally.
Strategic significance isn't always obvious. Rather than having boots on the ground in case the Soviets cross the Elbe, it's often more important that there's clear airspace to a potential hot zone and a large support cadre available to handle logistics. Rammstein AFB in Germany operates as a significant logistics hub for the current conflicts in the Middle East and Italy hosts quite a lot of our Mediterranean naval presence.
Some of the historical reasons for NATO are still current. Anti-ballistic missile placement vs Russia is an ongoing project.
The U.S. has military bases in many countries all over the world because it serves their security interests to have lots of large bases. It makes it near impossible logistically for a potential enemy to mount an attack on all bases, so there should always be a sizable amount troops for reinforcement/counterattack. They also serve as more convenient targets than the continental U.S. for attacks and aren't full of civilians so if one does get attacked its preferable to. Despite the cold war being over, the U.S. still isn't exactly friendly with Russia. These bases are also a military defense subsidy for the host country, a large army base that you aren't paying for upkeep on but contains soldiers will aid in your defense should you be attacked is a huge benefit. These bases are also partially responsible for the relative peace in Europe, its a lot harder for one country to attack another when they both have large amounts of soldiers from a third country that could take either side in a potential conflict. The U.S. has bases all over Europe because it is mutually beneficial for the U.S. and host countries.
The United States has Military bases where they are so that they can deploy troops to those areas faster and more easily.
Being able to deploy as quickly and easily as we are, is one of the major reasons why the US military is as powerful as it is. Having bases around the world makes the US military stronger.
isn't positioning an entire army corps deep in NATO territory (... Poland, Czech and Hungary...) a bit wasteful?
Not really, it's easier to attack Russia that way.
It is the legacy of the European nations refusing to pay for their own defence.
During the cold war various entreaties were made and accepted by the european states to have at least 5% of their GDP used to provide for their self defence. Only the UK, and I think maybe Greece, complied.
Due to this unwillingness the US was forced to retain credible forces of their own, both to strengthen their doctrine of deterrence, to provide a tripwire and in case of a purely conventional war. Much of the US conventional weapons budget went on equipment to thwart a mass armoured attack into Europe leaving them with legacy equipment that had a long shelf life.
This remains the case, where heavy tanks and massed artillery are of limited use in asymmetric warfare yet have to be based somewhere. Hence whilst artillery brigades dual-role as light infantry their heavy equipment still has to live somewhere, which in this case is the specially built and equipped european bases.
In short the American taxpayer picked up the bill whilst the european states concentrated on their economies.