According to the company's website, to qualify, you have to have purchased your original drink using the Starbucks app with a Starbucks card connected to your account. You'll have to present this to a barista to obtain a refill.
One other thing: if you leave the store, the deal's off.
(Just for fun, I'm using this photo of a Starbucks in Kyoto. I went there when I was in Japan last year, and it's by far the coolest Starbucks I've been to. See more photos here.)