Is it true that students from European countries have had many more advanced courses in mathematics when they arrive to Grad school in the US?
First and foremost, it is a important to note that there is a vast amount of variation between institutions and individuals, so generalizations are not terribly helpful here.

That said, historically there have indeed been differences in the philosophy and organization of European and American approaches to education. European education has tended to put more emphasis on knowledge, where American education has tended to put more emphasis on creativity and communication.  As a result, many European students are already specializing in high school, while many American students are required to remain generalists until university and may in fact still be taking classes outside their "focus" all the way through the end of their undergraduate studies.

This does not, however, provide any clear advantage to one group or the other when it comes to doing a Ph.D.  At the doctoral level, one needs both knowledge and also creativity and the ability to communicate well with colleagues. Thus, while students educated in one tradition are more likely to enter graduate school being able to prove Urysohn's Lemma, students educated in the other tradition are more likely to be able to effectively communicate why that proof is interesting.

In short: the word "appear" is in the original quote for a reason.