When you get down to its most basic structure, digital data is just a series of ones and zeroes, each known as a "bit." A Kindle, like most mobile devices and many modern laptops, stores that data on flash memory. This type of memory puts each bit on a transistor, which distinguishes between a one and a zero using trapped electrons. The more data, the more electrons are trapped. (It's important to note that the number of electrons doesn't change with more data, just the number of trapped electrons.)
Still, a trapped electron is in a higher energy state than an untrapped electron, and since Einstein's famous equation E = mc2 (didn't think an article about the Kindle would invoke Einstein, did you?) says that energy equals mass, a flash drive full of electrons in a higher energy state will be heavier. Of course, the difference is almost laughable: 10-18 grams, according to Dr. Kubiatowicz. That's a hundred-millionth as much as the difference between a full and dead battery, and because the most sensitive scales can only sense a difference of 10-9 grams, the extra heft is effectively unmeasurable.
Even still, it is a difference. When you pack your e-reader (or iPad, or smartphone, or laptop) full of files, you're ever so slightly weighing it down. Learn more about how digital memory works in the videos below.