When almost all Americans use the term, they mean abandoning extreme cultural practices that are illegal here, such as honor killings, hunting squirrels in the park for dinner, and so forth.
Americans do not generally expect "foreigners" to necessarily develop a love for crappy pizza, greasy hamburgers and American football. By assimilation, they mean things like being willing to live under the laws of this country. We don't kill women who lose their virginity before marriage, we don't tip walls over on homosexuals, and people are allowed to speak their minds, even if it offends someone - such as criticizing someone's religion or cultural practices.
That basic degree of assimilation is not too much to ask, and any more than that is not required of any American, whether native-born or an immigrant. It's any American's right to basically live their own life, as long as that life does not require preventing someone else from doing the same.
We can't maintain the country we have without some degree of assimilation, and respecting the Constitution and the great compromise it involves - you have your freedom so long as you grant freedom to others - should be one of the reasons why people want to live in this nation. Economic benefits will come and go, but the American tradition of living individuals to work out their own cultural icons is a great thing.
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. So says my mother in law!