For those on the Left, Zizek’s claims that capitalism is one of the (indirect) causes of the refugee crisis is not controversial. If you accept this premise, which Zizek’s audience typically does, the debate concerning the refugee crisis centers around what the (capitalist) West is obligated to do for the refugees.
For Zizek the answer is Marxist emancipatory politics (of course!). He is concerned that the direction the Left is headed is not only utopian, but it leads “the people,” the working class people, to turn to the far Right in order to address their problems, which takes the form of xenophobia.
In order to properly understand Zizek’s position, one should be aware of what he is rejecting: post-colonialism (which often coupled with multiculturalism and feminism). Post-colonialist thought rejects Eurocentric thinking, which has dominated Modern philosophy for the last 400 years. They often dismiss Western philosophy as a bunch of “dead white men.” Post-colonialists argue that the West, by pushing its values on the rest of the world, and using their values as a justification for bad behavior, has oppressed, dominated, and exploited other cultures, and failed to respect their autonomy.
And to be clear: Zizek is not anti-Muslim. He is against the Muslim fundamentalist Right, just as he rejects Christian fundamentalism.
Post-colonialism says, in order to avoid imperialist oppression of the refugees, the West is obligated to open its borders and let them move freely to the destination countries of their choosing. The West should tolerate their cultural differences. At its most extreme, post-colonialism may argue that female castration should be tolerated in minority populations because it is a part of their cultures (NOTE: this example is not explicitly mentioned by Zizek). Zizek rejects this line of thinking. If the refugees are in the West, we should crack down “brutally” (a favorite term of Zizek’s) on female castration, and call it what it is: female genital mutilation. The West must insist upon the enforcement of its values: freedom and equality. Specifically, the point of contention is on how one interprets the Western values of “freedom and equality.” Marxism holds that two important events, the Enlightenment and democratic capitalism in Europe, were necessary steps towards developing our Western values of freedom and equality. These events were unique to Europe. Consequently, Marx was inherently Eurocentric. The post-colonialist Left rejects the moral superiority of the West. Marxism insists upon it. You cannot have Marxism without moral superiority. (Actually, to be precise, Marxists dismiss “morality,” and are more concerned with real-world conflicts in which the oppressed insist upon respect and equality.) The greatest example Zizek gives of what the Left must reject tolerance of minority “cultures” that conflicts with Western freedom and equality: the child prostitution ring in England run by Muslims. The police failed to investigate out of fear of being labeled racist. If the Left cares about the poor and oppressed, it must be willing to set aside its taboos when their taboos protect the oppressors.
Secondly, Marxism says democracy taken to its extreme becomes unfreedom. There are those on the Left who argue that the refugees should be accepted by the West, and given citizenship, and even be allowed to vote so their voice will be represented. This would allow them to gain the political power necessary to protect their interests from being marginalized. This is the gold standard of democracy. Zizek is worried that if the refugees do not hold our Western values (i.e. they are Islamic fundamentalists), democracy will lead to unfreedom. For this reason we must limit democracy. Pure democracy destroys itself when the majority wants to oppress minorities (women, gays, etc). Zizek argues this is nothing new. Western liberalism already puts limits on democracy. For example: Angela Merkel, who, without democratic legitimization, chose to accept hundreds of thousands of refugees into Germany. The key is to conscientiously pursue the telos of a democracy for which individual freedoms and pure democracy should be sacrificed.
Zizek argues that Eurocentric emancipatory politics is more important than ever. There was a time when the introduction of capitalism into a country brought with it democracy. Today this is not necessarily the case. When Zizek mentions capitalism with “Asian values,” he is referring to a movement in the 90s in Asian countries rejecting Western values, specifically human rights and democracy, preferring one-party authoritarian governments, social harmony, and capitalist prosperity that gives preference to the well-being of the community. By trying to avoid harming other cultures, post-colonialism allows a new, more lean, less democratic form of capitalism to grow in power around the world.
Per Marx, the working class (proletariat) is the revolutionary class that capitalism brings to self-consciousness to stand up for itself (think Hegel). Marx did not call out the working class arbitrarily. But a failure of the Left to address their needs could easily push them to turn to the far Right. Elsewhere Zizek points out how the workers movement in Germany in the 20s (the socialist “Social Democratic Party” was the most powerful political party at the time) failed the workers. The workers did not turn to capitalism (liberalism) to solve their problems. They turned to the xenophobic Nazi party. Zizek is concerned that the events unfolding in Europe will lead to a repeat of the Nazis, and the refugee crisis is just one more straw on the camel’s back, and the present strategy of the Left will not stop it.