How do the Ural Mountains affect the distribution of climate regions in Russia?

The Ural Mountains have effect on the climate of Russia only partially, mostly in the mid-part of European Russia.

Explanation:

The Ural Mountains are a natural barrier for multiple things, and one of those things is the climate. Russia is a vast country but it doesn't really have lot of climate types, with only few dominating. These mountains are a barrier between Siberia and the lowlands on the western part of the country.

The Ural Mountains prevent in the summer the movement of the warm air masses from the western lowland of the country toward Siberia, thus making this part of Russia much warmer than it is supposed to be in accordance to its latitude. On the other hand, in the winter, the Ural Mountains manage partially to protect the western part of the country from the large, extremely cold fronts. This contributes to slightly higher temperature in winter in the European part of Russia, while in Siberia it contributes to one of the lowest temperatures in the world, only paired by Antarctica.