On the surface it seems like an odd line to become a catchphrase, but then even in context it's an odd line. It sticks out in a way that makes it easy to see why people would latch on to it. Plus it just sounds funny. The most interesting thing I think is that, out of place as the line might seem, it actually has a historical source:
By the way, according to a story in USA Today, director Paul Thomas Anderson derived the dialogue from “a transcript he found of the 1924 congressional hearings over the Teapot Dome scandal.” Sen. Albert Fall described oil drainage thus: “Sir, if you have a milkshake and I have a milkshake and my straw reaches across the room, I’ll end up drinking your milkshake.” He was convicted of taking bribes for oil rights on public lands.
So, there you go.* I'm sure if Albert Fall had made a slurping noise after he delivered this statement to Congress, it would have become a catchphrase in 1924 and maybe he wouldn't have gone to prison. Alas, Albert Fall was no Daniel Day Lewis and so flash forward to 2008 when Idrinkyourmilkshake.com has become the unnofficial destination on the web for discussing There Will Be Blood and Paramount Vantage (Blood's distributor) has been sending actual milkshakes to journalists and academy voters to keep them talking about the film in the run up to the Oscars.
In other news, There Will Be Blood is a fantastic film and if you haven't seen it you really should, if for no other reason than Daniel Day Lewis's amazing performance. You'll have to sit through three hours of movie to get to the "I drink your milkshake" line, but it's totally worth it. Just imagine, having actually seen the movie, you'll be able to quote the line with impunity. Wear that "I Drink Your Milkshake" t-shirt proudly, knowing that you're better than all those Johnny-come-lately internet poseurs.
* - Historical side note: The Teapot Dome Scandal revolved around Albert Fall, then Secretary of the Interior, taking bribes from oilmen Edward L. Doheny and Harry F. Sinclair (the historical inspirations for the Daniel Plainview character played by Daniel Day Lewis in There Will Be Blood) in exchange for no bid oil drilling contracts in Wyoming. If I had more time and/or energy I'd come up with a clever line about how, in 1924, Albert Fall went to prison for what Dick Cheney has done every day of his life and under three different Presidential administrations.