Games can most definitely be classified as art, but maybe not in the way most people think.
Each game presents a new medium upon which to paint one's own experiences; each is a different canvas evoking different emotions, feelings, or lack thereof (just as traditional art does). The differences are subtle: Halo is the acrylic where WoW is the oil, and yet you can paint whatever experience you want with all of these things, given the limitations of the material. You can't surf the Nā Pali coast in Halo any more than you can use traditional tempera paint to coat an acrylic gesso surface (not counting mods, of course).
Like each painting, sculpture, or art style, each game has a different meaning to different people, and enjoyment of each will depend on personal taste. How many refrigerators have been home to finger paintings over the years? And how many mothers have loved every messy line or wobbly resemblance of a smiling face? How many crappy garage bands have emerged from band practice feeling like kings of the world? Yet, regardless of the quality of these creations, we still refer to them, and love them, as art.
But the most important part to understand is that it's not only the game itself that is the piece of art, but the player's interaction with it; the art is channeled, and truly created, through use. A game with no player is a keyboard with no pianist; beautiful, yes, but incomplete. You can't appreciate the Mona Lisa if you have no eyes, just as you can't appreciate Super Mario Bros. without physically interacting with it. You can't appreciate Macbeth without experiencing it (reading, watching, or listening), just like you can't appreciate the aforementioned Halo without needling some grunts. In order to appreciate an art form it takes a certain degree of fluency; if you don't speak English then Shakespeare may be lost on you, just as if you don't learn to use a controller then you won't be able to fully experience a game in the way it was intended.
But wait, can action or interaction really be classified as art? Certainly. Look no further than the art of dance or stage play. We can watch a dancer, just as we can watch other people play games, but a type of magic happens within the dancer that can at best be empathized by the outside observer and at worst misunderstood.
Roget's 21st Century Dictionary defines art as "creation meant to communicate or appeal to senses or mind." There's no arguing that video games fall under this definition. It's easy for a non-game player to disregard games as art, for you can't understand something without using it in the way in which it was intended.
If you are reading this, you've probably had a video game experience that moved you in a way that no piece of traditional art ever could. Think back upon an especially exhilarating or captivating moment. Do you realize that that amazing moment was created through the interaction between yourself and the game? And thus I say to you: welcome, artist!