The distinctive shape of the martini glass truly serves one purpose: to look amazing. There are some side benefits of a good-looking glass, though.
The stem is longer than you'll find on most coupes or other wineglasses, which keeps the heat of your hand far from the drink itself. Martinis are "shaken, not stirred," as James Bond insisted, with ice, and you want the vodka or gin and vermouth to stay cold as long as possible.
You can, of course, stir the ingredients with ice and then strain the cocktail into the glass, but if you can shake it like Polaroid picture, why wouldn't you?
The wide-open rim also allows more of the drink to come into contact with the air so it can "breathe," allowing its flavors to open up. Many other cocktail and wineglasses curve in at the rim, at least a little, reducing that aeration effect.
There's also a rumor floating that back during Prohibition, it was easier to toss the contents of a martini glass during raids on speakeasies. But if you toss a drink yourself, you'll realize how fast you can get rid of just about any liquid from any glass when you're trying not to get caught by the police.