> I should imagine the closest language to Russian would be Belorussian...
Yes... But I learned this from lecture above right now.
As a russian I cannot understand fluent speech of both of them. Some words are similar, some ideas are understandable, but fluent flow of speech quickly becomes meaningless.

> How came Belorussian to evolve as an independent language
Language is very flexible construction. There are always two process accross territory: morphing into something new and melting into something similar.
You should understand that so called "old russian (slavic) language" is as much distant from modern russian, as modern russian is distant from ukranian. Moreover there were many dialects in past too.
All three modern eastern slavic languages are far from their ancient roots also. They are melted and divided from these roots by historical/political reasons.
As said in the lecture above, great role in separation of modern ukranian and belorussian from russian belongs to Grand Duchy of Lithuania. And so on.