I will try to explain it as clearly and succinctly as I can. I will explain the simplest version of the puzzle, the 'monoglot' version. We shall cleave to English as to our dear mother.
The puzzle is generated by the collision of two principles, one concerning reference, the other concerning disquotation. Call them MILL and DISQ.
MILL: The reference of a proper name is direct: not routed through sense as in Frege. The meaning of a name is exhausted by its reference. The semantic value of a name is just the object to which it refers. (Gareth Evans plausibly recommends 'semantic value' as the best translation of Frege's Bedeutung.)
DISQ: If a normal English speaker S sincerely assents, upon reflection, to 'p,' and 'p' is a sentence in English free of indexical elements, pronominal devices, and ambiguities, then S believes that p.
The puzzle is interesting, and not easily solved, because there are good reasons for accepting both principles. The puzzle is puzzling because the collision of the two principles takes the form of a flat-out logical contradiction.
And as we all know, philosophers, while they love paradoxes, hate contradictions.
(DISQ) strikes this philosopher as a principle than which no more luminous can be conceived. How could one who is competent in English and familiar with current events sincerely and reflectively assent to 'Hillary is a liar' and not believe that Hillary is a liar? The intellectual luminosity of (MILL), however, leaves something to be desired. And yet it is plausible, and to many experts, extremely plausible. Brevity being the soul of blog, I cannot now trot out the arguments in support of (MILL).
The collision of (MILL) and (DISQ) occurs at the intersection of Mind and World. It comes about like this. S may assent to
a. Cicero was a Roman
while failing to assent to
b. Tully was a Roman
even though
c. Cicero = Tully.
Given (DISQ), S believes that Cicero was a Roman, but may or may not believe that Tully was a Roman. But how is this possible given the truth of (c)? Given (c), there is no semantic difference between (a) and (b): the predicates are the same, and the names are semantically the same under (MILL). For on the latter principle, the meaning of a name is its referent. So sameness of referent entails sameness of meaning, which is to say: the semantic content of (a) and (b) is the same given the truth of (c).
How can S believe that Cicero was a Roman while neither believing nor disbelieving that Tully was a Roman when the sentences express the very same proposition? This is (an instance of) the puzzle. Here is another form of it. Suppose S assents to (a) but also assents to
d. Tully was not a Roman.
On (DISQ), S believes that Tully is not a Roman. So S believes both that Cicero was a Roman and that Tully was not a Roman. But Cicero = Tully. Therefore, S believes that Cicero was a Roman and S believes that Cicero was not a Roman. This certainly looks like a contradiction.
It seems that our governing principles, (MILL) and (DISQ), when applied to an ordinary example, generate a contradiction, the worst sort of intellectual collision one can have.
The Paderewski case is similar. On different occasions, Peter assents to 'Paderewski is musical' and 'Paderewski is not musical.' He has no qualms about assenting to both since he supposes that this is a case of two men with the same name. But in reality he is referring to one and the same man. By (DISQ), Peter believes both that Paderewski is musical and that Paderewski is not musical. Given (MILL), Peter believes contradictory propositions. How is this possible given that Peter is rational?
Given the luminosity of (DISQ), one might think the solution to Kripke's puzzle about belief is simply to jettison (MILL).
Not so fast. There are powerful arguments for (MILL).
