Coming from an American, Its not that we dislike Europeans, we just don't want to be Europeans. If you've ever heard of American Exceptionalism, that's what I'm talking about. Americans just enjoy being different. We call football soccer, and we don't really enjoy the game, we use or own measurement system, etc...
As for us, "mostly being European" that argument hasn't been valid since the 1800s. Yes, many Americans are descended from Europeans. Does that make them European? I think not, since they have no knowledge of European culture, history, etc. Hell, the vast majority have never even been to Europe, myself included.
The reason Americans tell immigrants to go back to their countries is actually far more specific than you think. Its usually targetted at the illegal Mexican immigrants. America has had two huge waves of immigration in the past, I believe the 1st was in like the mid 1800s,and the 2nd was in the early 1900s. Now, some say we are experiancing our third wave. The only difference is, they aren't coming across the Atlantic this time.
During the first two waves of immigration, people were telling those immigrants to leave too. During the 1900s, you'd be hard pressed to find any American who had sympathy for an Irishman, or any one from Poland. Now, the b hate has shifted towards the Mexicans, but it will pass.
As for the reason, well, people have to now compete for blue collar jobs, and many Americans don't want that. My personal opinions on that matter are kinda detailed, but I think this post is long enough, so I'll leave it at that.