Kripke aims to argue "that the considerations in Donnellan's paper, by themselves, do not refute Russell's theory" of descriptions. He remains officially neutral on whether Russell's theory is correct.
Kripke's early discussion, in section 3, presses on Donnellan's unclariy, towards the end of the paper, about (i) what "statement" someone makes when they use a description referentially and (ii) exactly what kind of distinction the referential–attributive distinction is. Kripke insists that if the distinction is merely "pragmatic", and is not any sort of ambiguity, then it is hard to see why Russell should be concerned. He meant to be analyzing the meaning of descriptive statements, not "uses" one might make of them.
The key to Kripke's analysis of the phenomenon Donnellan uses to motive the referential–attributive distinction (henceforth, the "D-phenomenon") is the distinction he draws on p. 111 between speaker's reference and semantic reference. The distinction is inspired by, and is arguably a special case of, Grice's distinction between what one says when one makes a given utterance and what one means, which Kripke briefly explains. The semantic reference of an expression is fixed by linguistic facts or conventions and (ignoring things like tense and demonstratives, for the moment) does not vary from occasion to occasion. The speaker's reference, on the other hand, can vary: It is "that object which the speaker wishes to talk about, on a given occasion, and believes fulfills the conditions for being the semantic referent of the designator" (p. 111).
It is tempting to borrow from Grice and say that the speaker's referent is the object one means. It would be worth exploring whether some such equation can be made to work, i.e., to try to show as precisely as possible how Kripke's distinction can be seen as a special case of Grice's. (This suggestion is only intended for students who have some familiarity with Grice's distinction.)
Kripke suggests further that the reason speaker's reference can come apart from semantic reference is because one can use a name with two sorts of intentions. On the one hand, one can use the name with the sole intention of referring to its semantic reference. In that case, the speaker's reference is guaranteed to be the same as the semantic reference. On the other hand, however, one can intend to refer to some person, say, that one can see, and thinking that this person is NN, one might then go on to use "NN" in saying something about them. If one is wrong about who that person is, them the speaker's reference will not be the same as the semantic reference, but will be the person one can see.
One of Grice's "conversational maxims" is the so-called "maxim of quality", which says that one should only say what one has reason to believe is true. Let's accept this as a general (but not strict) rule of conversation. Then if someone says "The F is G", we may ask what sorts of reasons they might have to believe what they are saying. Broadly speaking, those reasons divide into two sorts. One might have general reasons to believe that the F is G: One might think anything that is F is G and also think that there is exactly one F. Or one might have particular reasons: One might think that some particular thing x is G and also think that x is the one and only F. How might this observation be used to reinforce Kripke's point?
Perhaps one of the most famous features of this paper is the "test" that Kripke describes on p. 113:
If someone alleges that a certain linguistic phenomenon in English is a counterexample to a given analysis, consider a hypothetical language which (as much as possible) is like English except that the analysis is stipulated to be correct. Imagine such a hypothetical language introduced into a community and spoken by it. If the phenomenon in question would still arise in a community that spoke such a hypothetical language (which may not be English), then the fact that it arises in English cannot disprove the hypothesis that the analysis is correct for English.
Kripke then introduces three "Russell languages", in which various stipulations are made about how descriptions behave semantically (pp. 113-4). He goes on to argue that the D-phenomenon would arise in all three of those languages.
Make sure you understand why Kripke's test is supposed to be a good one and how it is supposed to apply to the referential–attributive distinction.
Kripke allows that it is possible that descriptions behave ambiguously, and he introduces the D-languages to illustrate what this hypothesis would involve. He then argues on very general, methodological grounds that we should prefer the hypothesis that English is a Russell language.
In some ways, Kripke's entire case against Donnellan is contained in the example he gives on p. 111 about Smith and Jones. What this example purports to show is that the D-phenomenon has nothing special to do with descriptions, but can also arise with proper names. Why is it such a problem for Donnellan if that is true?
Kripke does not apply his "test" to what he calls the "unambiguous D-language". If we did, I claim, we would find that the D-phenomenon arises within the unambiguous D-language itself. I claim further that this is a much more powerful reason than the ones Kripke gives to think that English is not a D-language. See if you can spell out this argument.