Learn Italian: Se ne parla costantemente, ma ce ne si stanca. - People constantly talk about it, but one gets tired of it.

When the impersonal pronoun si is followed by another pronoun starting with a consonant like n or l (such as ne, lo, or la), its vowel changes from i to e. Therefore, si plus ne becomes se ne.

In ce ne si stanca, where does the ce come from?

It comes from the impersonal form of the reflexive verb stancarsi (to get tired). To avoid the awkward repetition of si si (impersonal si plus reflexive si), Italian changes the first pronoun to ci, making it ci si stanca (one gets tired). When you add ne into the mix, ci changes to ce before ne, resulting in the cluster ce ne si.

What exactly is the word ne replacing in this specific sentence?

In both clauses, ne replaces a phrase starting with the preposition di. First, it replaces what is being talked about (parlare di means to talk about). Then, it replaces what people are getting tired of (stancarsi di means to get tired of).

Could se ne parla also mean "he talks to himself about it"?

Grammatically, yes. Because si can also be a third-person reflexive pronoun, se ne parla could technically mean "he talks to himself about it". However, this is quite rare in practice, and Italian speakers will almost always understand this combination as the impersonal "people talk about it" or "it is talked about".