It's because storing new data involves holding electrons in a fixed place in the device's memory. Although the electrons were already present, keeping them still rather than allowing them to float around takes up extra energy- about a billionth of a microjoule per bit of data.
Using Einstein's E=mc² formula, which states that energy and mass are directly related, it's been calculated that filling a 4GB Kindle to its storage limit would increase its weight by a billionth of a billionth of a gram, or 0.000000000000000001g. This is roughly equivalent to the weight of a small virus, while the equivalent number of books (about 3 500) would weigh approximately two tons. :o)