The Wikipedia page on Russell's paradox states
if $R$ were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if $R$ were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal
We have to assume that there are such things as extraordinary or "abnormal" sets for this paradox to be valid. The solution to the paradox is to change the definition of a set so that it cannot include self-referent collections.
Why do we assume that sets must be normal or abnormal in the first place? To my eye this whole thing can be avoided if we do away with what appears to be an unnecessary assumption.
Your quote seems to be taken from Wikipedia page Russell's paradox. Specifically from the section titled Informal presentation.
Which is exactly what this is: an informal presentation. Which begins by setting an ad-hoc terminology of what is a normal set, and any set which is not normal will be called abnormal. But it is important to emphasize that this is an ad-hoc terminology for the sake of explanation. This is like me saying "let's call a person who can type single-handedly with their non-dominant hand groovy and otherwise they will be called broody", we have created an artificial property of humans, it's not something intrinsic to being a human being, but every human being is either groovy or broody.
But by the law of excluded middle, given a set, it has to be one of the two: normal, and otherwise abnormal. And this is another important point, given a set, it either satisfies the property of "being normal", or it doesn't, in which case it is "abnormal".
This is the source of the paradox. If the set $R$ is normal, then it has to be abnormal; and if it is abnormal, then it will have to be normal. Since a set is never both normal and abnormal, it can only be that $R$ is not a set to begin with.