Trying to understand the history behind Russell's Paradox

[GrandizerII]

So I'm reading about Russell's Paradox on IEP, and I'm trying to fully grasp the History section as best I can but currently struggling.

Link to article:
https://www.iep.utm.edu/par-russ/

The second paragraph is what I'm currently stuck at. Here's the paragraph in full:

Russell discovered the contradiction from considering Cantor's power class theorem: the mathematical result that the number of entities in a certain domain is always smaller than the number of subclasses of those entities. Certainly, there must be at least as many subclasses of entities in the domain as there are entities in the domain given that for each entity, one subclass will be the class containing only that entity. However, Cantor proved that there also cannot be the same number of entities as there are subclasses. If there were the same number, there would have to be a 1-1 function f mapping entities in the domain on to subclasses of entities in the domain. However, this can be proven to be impossible. Some entities in the domain would be mapped by f on to subclasses that contain them, whereas others may not. However, consider the subclass of entities in the domain that are not in the subclasses on to which f maps them. This is itself a subclass of entities of the domain, and thus, f would have to map it on to some particular entity in the domain. The problem is that then the question arises as to whether this entity is in the subclass on to which f maps it. Given the subclass in question, it does just in case it does not. The Russell paradox of classes can in effect be seen as an instance of this line of reasoning, only simplified. Are there more classes or subclasses of classes? It would seem that there would have to be more classes, since all subclasses of classes are themselves classes. But if Cantor's theorem is correct, there would have to be more subclasses. Russell considered the simple mapping of classes onto themselves, and invoked the Cantorian approach of considering the class of all those entities that are not in the classes onto which they are mapped. Given Russell's mapping, this becomes the class of all classes not in themselves.

I understand the first few sentences of the above paragraph just fine. The first sentence I'm stuck at trying to understand is the following:

Some entities in the domain would be mapped by f on to subclasses that contain them, whereas others may not.

Can anyone explain what the author means here, especially the last part of that sentence? What "others" may not be mapped by f on to subclasses that contain them? How in this context can an entity be mapped to a subclass not containing it? Or am I misunderstanding the term "map" here?

I want to try to understand this bit by bit so I can fully understand the whole paragraph. Thanks.