The main language spoken in Siberia is Russian. That is a direct result of the expansion eastward of the Russian people during the reign of the czars. That expansion involved a policy of "russification," meaning the government in Moscow followed its expansion to the Pacific Ocean with programs to populate the newly conquered territories with ethnic Russian people. An integral component of russification involved compulsory use of the Russian language.
While Russian is the main language used in Siberia, that vast region is home to many non-Russian -- and many non-Russian-speaking -- ethnic groups. These include Turkic speaking peoples, dialects specific to the region of the Urals (the mountain chain constituting the division between European and Asian Russia), Mongolic peoples, and many more. Surviving speakers of these languages range in number from a few hundred to hundreds of thousands.
Because Siberia, and Russian Far East, which is often included in discussions of Siberia, borders so many other countries, and because borders are political constructs often drawn without regard to ethnic unity, small numbers of other groups can be found that still use their native languages. These languages include Korean and Chinese dialects.
The main language spoken in Siberia is Russian, the official language of Russia. However, there are a lot of people who speak a variety of native languages, other than Russian.