Why did russia build the trans-Siberian railway?

Much the same reason as any country built its trunk railway routes. It gives relatively rapid, efficient transport of goods and passengers along the length of the line, which has various branches along it, between the capital and the Baltic and Pacific ports of this vast country. Such transport is far quicker than going by sea, though there is a busy shipping-lane right along the North coast of Siberia. It is also easier to keep a railway open in the Siberian Winter, than a frozen sea-lane.

It carries a huge amount of freight, including containers between Europe and the Far East; timber, agricultural produce and minerals from within the country, and so on.