The definition of "vintage" is a little different for fashion than it is for a fine wine -- think more like vintage cars or a vintage guitar. Vintage clothing is clothing that is more than a decade old, and, like a '52 Meteor or a '65 Corvette, it usually has a particular style that ties it to its time period. Girls like wearing vintage fashions for three reasons: often they fit better than modern clothes, they offer a wider variety of styles, and they're fun to shop for. All of these things are potentially true for guys as well.
Since "vintage" is defined as "anything older than your ten-year-old nephew," clothing from a huge number of different eras falls under the heading. Modern clothes tend to be designed to fit particular body types -- the types that are "in" right now. Vintage clothes are designed to fit the types that were "in" back when they were made. That means that skirts from the 1940s make the hour-glass girl look amazing, and girls without much in the way of hips or waists look awesome in dresses from the 1920s. The same principles apply for men: if you have the broad shoulders and slightly narrower hips that were the ideal during World War I, a military-cut jacket from the time will show them off better than anything designed for the modern, skinny-jeaned hipster.
A related but separate idea is that the variety of styles available to the vintage shopper fits not just more body types, but also more personality types. So you're not interested in wearing maroon pants, a checked shirt and a fedora. That doesn't mean you have no options but baggy t-shirts and jeans. If modern styles don't suit you, there's almost certainly some vintage era that does.
Girls also like to wear vintage clothing because shopping in the little hole-in-the-wall boutiques with racks of random fashion items from the last hundred years is fun for them. They like exploring the shelves of unusual styles, looking for the top or the dress or the pair of shoes that makes them look models and feel like they're showing off the uniqueness of their personalities. It may go against the male grain to enjoy shopping for clothes, but vintage stores at least increase the likelihood of finding something you feel great wearing. If that's not enough to turn vintage shopping into a hobby, it may still make the experience a little less painful than shopping in department stores.