Who proved it? When?
Georg Frobenius, Über die Primfactoren der Gruppendeterminante, Sitzungsber. Akad. Berlin (1896) 1343-1382. The theorem is announced at the beginning, p. 1344:
and proved at the very end, p. 1382:
P.S.: As to an agreed-upon name for this theorem, I think there isn't one (or googling would reveal it). Too much competition from other results of Frobenius! So, other than in the book by Etingof et al. (who innovate by calling it "Frobenius divisibility theorem"), one finds mainly turns of phrase like "Frobenius's theorem that the degrees of the irreducible characters divide the order of the group", as in Curtis (who devotes two pages to this theorem and Burnside's second proof of it).
An interesting byproduct of this archaeology is some light shed on the words "degree" and "irreducible" in representation theory — namely, they initially qualified factors of a polynomial, the group determinant.