There are a lot of answers to the question. It depends on the group you are talking about given that Slavic covers a huge region.
Primarily it is just ignorance and conditioning. Americans have been taught that all of eastern Europe is communist and therefore the enemy. In fact they are not all anything. Even when you are talking about the old concept of the USSR the real problem, and the leaders of both sides greatest nightmare, was that people don't know each other.
We are far more alike than different. In fact most people around the world want the same things. Right now we are being drilled that China and therefore Chinese are all plotting against us and hate our very existence. Well I have lived in China and I know a fare number of Russians as well as other Slavic' and all I can say is the day the common people meet is going to be one heck of a party, rather than a war.
But still there are people who are evil Romas, Gypsies, have a much earned reputation for being thieves and lowlifes, Chechnyans have earned their ire as well. Right now the enemy is Muslims, more specifically Islamics. But how many Americans know the history behind the hate?
But we have, over time, felt the same way about the Italians, the Irish and any number of others.