Why do Republicans hate trains? Is this really just a situation in which Democrats and Europeans find value in trains, so therefore it is un-American to ride trains?
This is part of the reason why yes, but conservatives here have long had a bitter hatred towards both passenger rail and public transportation in general. (Hard to notice, I know, with all of the other people and things they bitterly hate.)
What it comes down to I think is that their understanding of society is basically feudal. Today's right wing considers even the most basic anthropological definition of society; cooperation and loyalty to strangers, and acknowledging some level of dependence to them, to be socialist. People exist for the purpose of competing against each other for dominance, and will only grudgingly band together with members of their own tribes to fight off a common threat to their dominance. There is no common duty, no concept of 'us' existing beyond opposition to common enemies and paying homage to totems of state power such as the flag or the right-wings cartoon version of the constitution.
Their main problem with trains and public transport then is that it's all so damned public. It requires being in at least the physical presence of social outsiders, people from different backgrounds living different lives; often the sort of people who the right-winger bases a huge chunk of his own self-worth on looking down upon. People without cars, which is a deprivation akin to homelessness in the conservative mind, city dwellers, who are either impoverished brown-skinned factory and jail fodder who can't afford to own a McMansion like Real Americans do or some filthy hipster/homosexual who could afford a real home if they wanted to but choose the city only to better indulge their pagan depravity.
Driving, on the other hand, is either done alone or with people from one's own family or social group. Our auto based transport system is a fantastic way to keep the separate classes of America separate, which is of course a big part of the reason for why we were made to get around this way; and there are quite a few among us who won't broke any challenge to this separation and the personal sense of superiority that this separation makes possible.