Compare with your previous post regarding On Denoting for Russell's approach to "denoting phrases": Russell's theory is based on the idea of not considering locutions like "the father of Charles II" as names but as complex formulas.
The example regarding the "unit class" is used by Russell in the context of the discussion of Frege's theory (that Russell want to adopt as much as possible) that "every sentence has both a meaning and a denotation":
Or again consider such a proposition as the following: "If u is a class which has only one member, then that one member is a member of u", or as we may state it, "If u is a unit class, the u is a u". This proposition ought to be always true, since the conclusion is true whenever the hypothesis is true. But "the u" is a denoting phrase, and it is the denotation, not the meaning, that is said to be a u. Now if u is not a unit class, "the u" seems to denote nothing; hence our proposition would seem to become nonsense as soon as u is not a unit class.
So, Russell's point of view is the following: he does not reject the (now common) truth-functional analysis of "if..., then..." conditionals (implying that a conditional sentence is true when the antecedent is false) but he is saying that if we want also to follow the "common sense" approach that asserting something about a non-existing thing (or not well-defined one) is simply false and not meaningless (nonsense), we have to find a way to analyze phrases like "the u" in such a way that they have a truth value also when the uniqueness part (expressed by "the") is not satisfied.
The example can be rewritten using the phrase "the satellite of Earth is the Moon" where "satellite of Earth" is a predicate applying only to one object; thus the set M = { x ∣ Earth's sat(x) } is a singleton and we may freely say "the M" and use a proper name Moon to denote it.
The set J = { x ∣ Jupiter's sat(x) } instead is not a singleton and thus we cannot write "the J" ("the satellite of Jupiter") and call it e.g. Io because Jupiter has 95 satellites and Io is only one of many.
In conclusion, Russell want to avoid that statements containing "denoting phrases" will end up "nonsensical", i.e. neither true nor false:
... difficulties which seem unavoidable if we regard denoting phrases as standing for genuine constituents of the propositions in whose verbal expressions they occur. Of the possible theories which admit such constituents the simplest is that of Meinong. This theory regards any grammatically correct denoting phrase as standing for an object. Thus "the present King of France", "the round square", etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law of contradiction. It is contended, for example, that the existent present King of France exists, and also does not exist; that the round square is round, and also not round, etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred.
This is the complete context: if "the u" is considered a name we can use it as a subject in a sentence and then what happens with the conditional:
"If u is a unit class, the u is a u"?
When considered from the "pure propositional" point of view, when the antecedent is true, the consequent is also true ("the M" is the (only) member of the set M and thus "the M is an M"), while when the antecedent is false the conditional is trivially true.
Thus, the complete statement is always true (a logical law).
But if we adopt a "finer" approach, that of predicate logic, when we use "the u" as a name (a subject of predication), when there is no denotation we get a sentence that can be both true and false (compare with the example: "The round square is round and a square").
In this case "the u" is (by construction) the unique element of the set u, and thus it must be unique, and is the (unique) element of the set u, and thus it is not unique, because u not a singleton.
