The advertisement is a play on a popular folksy phrase with many variations along the lines of:
You can take the [girl] out of the [country], but you can't take the [country] out of the [girl].
As just one example, the intended meaning is that a "country girl" (as in Southern/Midwestern American girl) could move to the "Big City" but still live and act in a "country" manner.
Essentially the idea of the saying is that you can change your locale and location, but your essence remains the same. It is sometimes meant to be offensive, such as in the example above the speaker could be suggesting that the "country girl" is unrefined and cannot succeed by the standards of the "Big City" life.
Or, it could actually be taken as a statement of self-empowerment -- suggesting in fact that no matter where you are or what your circumstances, you are still true to your roots and your core being.
Therefore, in the case of this advertisement, Skyy is suggesting that the best vodka comes from Russia, such that Russia itself is a metaphorical component of the essence of vodka -- to the point that you cannot separate vodka from the notion of Russia. This takes Russia, the country, into a metaphorical existence as some essential component of something. Thus the reasoning behind the definite article with "the Russia".
The advertisement is simply a joke to highlight that the while there is an innate harshness of grain-based alcohols -- since they retain a high percentage-by-volume (ABV, alcohol by volume) of ethanol (alcohol) -- there is also a desirability for taste and the notion of palatable "smoothness". So, expensive liquors are often touted for their "smoothness" as seen as a product of filtration, and many liquor brands will advertise something like "filtered four times", or some such similar statement. More filtration meaning less harshness in taste.
And to fully beat this dead-horse on explaining the joke, the idea is that Skyy has created a vodka so smooth by filtering it so well they even removed the essential "Russia"-ness of it, something assumed entirely inherent. And, again, their language in the wording of the advertisement is a direct reference to the colloquial, folksy phrase.