All of this answer is a discussion of Schmidt's "Crisis of Parliamentary Democracy" where that quote originates from. For Schmidt, government is fundamentally about who gets to decide. This is exemplified by the judicial system, where citizens literally give up their own liberty to ask the government to solve their problems.
Liberalism asks to accept that all people have fundamental rights, but a democracy requires that we separate between citizens and non-citizens. Citizens and non-citizens are not-equal. A democracy must be able to treat them differently and (when necessary) eliminate those people who are hazardous to its citizenry. This makes democracy fundamentally incompatible with liberalism.
Liberal democracy requires equality. One common idea in America (and maybe other places) we have a general conception of equality based on our existence as humans ("all men are created equal", etc).
A full discussion of why equality is necessary for a democracy can be found in Political Theology (another work by Schmidt), but the short story is that in order for the law to successfully govern it needs a "medium of homogeneity". When people aren't consistent, the law can't be applied consistently and good government is impossible.
There's a problem with the kind of general equality: it isn't enough. Schmidt calls this "liberal equality". The problem is that it's too abstract to be effective. When a court is attempting to apply the law to a case, a philosophic human equality is too abstract to effectively resolve a problem. Democracy therefore requires a more specific kind of equality - the equality of citizens. Citizens can be ascribed rights and privileges through the law which can be withheld from non-citizens.
This is the basis of the first part of the quote. Democracy requires homogeneity between citizens. The liberal idea of general human equality isn't enough, democracy requires real ("substantive") social sameness between the community of citizens. Otherwise, it can't govern effectively.
Citizens are expected to be relatively similar. Not only a legal kind of equality, but "substantive homogeneity". That is, they should be expected to be a relatively uniform community. That could mean sharing a language, having similar values, a common understanding of their government, etc.
Non-citizens may not share all these traits. They are outside the community. When a non-citizen becomes a member of the community, they must assimilate. Think of the "melting pot" mythology in the United States. It's a view of foreigners assimilating to a fictional United States culture which serves to keep the country relatively homogeneous.
However, the state must also be free to destroy heterogeneous elements. This is partially a survival strategy: since democracy requires homogeneity to function, it must be able to defend itself from heterogeneity.
Schmidt was un-apologetically one of the philosophic leaders of National Socialism in Germany. It's easy to see where he goes from here: the state (even a democracy) must exercise whatever power necessary to free itself from foreign social elements.
This is the core of the second part of the quote.
The Democratic Paradox. Chantal Mouffe.
Carl Schmidt article on Stanford Encyclopedia of Philosophy.