According to this message on JewishGen, a state rabbi was a requirement in Russia. To hold the position, candidates had to known Russian. The state rabbi or "Kazeny Ravin" in Russian had bureaucratic duties:
"he had to register births, marriages and deaths, keep these
vital registers and issue birth, death and marriage certificates."
Each town with a Jewish population had to elect a state rabbi. The people who participated in these elections were called Rabbi's Electors. These electors were comprised of the heads of the households in the town along with the board of directors of the local shul (see here, pg. 181, n. 195).
Note: If I'm not mistaken, Rabbi Prof. Shnayer Leiman once said that Rabbi Yudel Rosenberg, author of the famous (or infamous?) Nifla'ot Hamaharal (the main source for the myth of the Maharal's golem) had at some point in his life studied Russian so he could be elected as a state rabbi.