A few years back my grandfather made some wine out of potatos, he said it tasted pretty good. How come there's never been a mass international potato wine industry?
Before I dive into the answer lets look at what is needed to produce an alcoholic beverage:
It turns out that the European wine grape (vitis vinifera) is extremely talented at providing these three characteristics. Of all fruits it has both the highest water content and the highest sugar content. The high water content makes it extremely easy to crush and release a high percentage sugar solution. Grapes also harbor many different species of yeast with some of them being highly talented at surviving in a high alcohol environment.
For these reasons it is extremely easy to make wine. Even by accident. Its probable that the first alcoholic beverages were wine based with a grain/grape/honey/herb mix following soon afterwards in the timeline.
Aside from having physical characteristics that make alcohol production easy, another extremely important facet of vitis vinifera is that it tastes very good as wine. Of course this is somewhat subjective, but if you try almost any other type of fruit that is fermented to dryness (all sugar fermented to alcohol) you will notice that they tend to taste quite austere and bitter/sour. Of course many people enjoy traditional ciders and meads as an acquired taste, but wine made from vitis vinifera has a seemingly unique characteristic in that it can still taste fruity sweet even after being fermented dry. Even other species of grapes lack this characteristic that vitis vinifera has. This fruitiness that balances out the acidic nature in fermented fruits has historically made wine from these grapes very sought after.
Lastly, since vitis vinifera wine easily ends up with a quite high natural alcoholic content it keeps better than pretty much any other non-distilled alcoholic beverage. Mead can rival it, but mead lacks the innate fruity sweetness that humans find so appealing in wine.
I would love to know how rice wine (sake) has become so popular, and still manages to retain an inherently sweet taste with such a high alcohol content?
If you are interested in delving deeper into the world of wine there is a book called, Windows on the World Complete Wine Course, by Kevin Zraly. Its long but easy to read and is a great overview of wine.
In addition to Qweniden's answer, what we might refer to as marketing was also a huge help. Wine, particularly champagne, benefited from being a symbol of wealth and as standard of living rose everyone wanted their piece of the pie, what's better than something portable, intoxicating and with a relatively long shelf-life?
Also, lets not forget that wine is necessary for Catholic communion, where Catholicism went, wine tended to follow! Beyond this, wine, much like wheat, is deeply symbolic in Western culture. It's tied to the Greek symposium, Rome, etc. and has long been a symbol of plenty.
Should you be interested in the marketing aspect you might want to give Kolleen Guy's "When Champagne Became French" a look.
For its role in Catholicism, it pops up fairly frequently in work and religion and empire, off the top of my head I know Jeffrey Pilcher writes about it in "Que vivan los tamales!" and the other stuff is mentioned in Janet Chrzan's "Alcohol: Social Drinking in Cultural Context" which is a great read.