Why did Russia want to expand eastwards, and what where the benefits?
Russia expanded (mostly) eastwards for the same reason Britain expanded over seas: the path of least resistance.
On the East there were rapidly decaying but still meddlesome remnants (Kazan, Turan, Astrakhan) of the Golden Horde. Obviously this is where one would apply the force: relatively easily secure the Eastern border and only then concentrate on the West and South.
Of course, Russia was, in fact, expanding in all available directions, wherever there was a weak spot in the neighborhood. It's just that there was much more space in the East and resistance there was much weaker.