When Appiah says “cultural imperialism” he means when a culture begins to dominate another culture, replacing their customs and ways of life with new, different ones. An example of this would be Levi’s, who are a western company that have spread to every continent. Many cultures have abandoned their traditional clothes to wear these cheaper, modern, western clothes instead.

A Chinese man wearing jeans waits for his food.
What Appiah says is the “golden rule of cosmopolitanism” is, before translation, “Homo sum: humani nil a me alienum puto.” When translated, this means “I am human: nothing human is alien to me.” What I believe Appiah is trying to say in this is that, as a human, you must learn to accept the ways of others, whether you agree with them or not.
What Appiah means by the statement “cultural purity is an oxymoron” is that there is no single culture, no matter how far back you go, that has not met with another for one reason or another. For example, even centuries ago people were getting the materials for their traditional clothes imported from other countries and even other continents.