What is the role of "virtually" in this sentence?

You're over-thinking it. Virtual here means "in essence, but not in fact".

to clarify:

I certainly don't even understand the difference between the virtual extinction of the buffalo and the extinction of the buffalo.

Extinct means "no longer in existence; having died out." For the buffalo to be extinct, every last buffalo must have perished from the Earth.

The North American buffalo is not extinct. There are a few thousand of them left. They are virtually extinct.

It is virtually impossible to escape this place!

It's very, very difficult to escape, nigh on impossible, but nothing is completely impossible. Virtually nothing...

to clarify some more: I think virtually can be distinguished from almost, in that almost merely denotes quantitative proximity. If a bullet whizzed by your head, it almost killed you -- but there's a tremendous difference between being almost killed and actually being killed.

Virtually means the difference between the current state and the named state is detectable, but not significant. The buffalo is virtually extinct in the sense that it exists only in a few scattered domesticated herds. The continental masses of bison that once ruled the Great Plains, their essential buffalo-ness, are gone forever.

In these contexts, virtually means "almost".

It is virtually [almost] impossible to escape this place!

and

It is impossible to escape this place!

are different. The first implies that it is possible, although very difficult. It is the same with your extinct sentence. If an animal is extinct, there are none left. If it is virtually, or almost, extinct, there are very few left and it will be extinct soon, unless something is done to prevent its decline.