Here's my explanation:

Let's say you're a company that's providing an internet connection to a business. The company trusts you, so there's only compression of bits over the wire, not encryption, and you're aware of the compression scheme the company is using to send their bits to you. You're charging the company a premium for using the line you manage but you also lease the line, so it's in your interest to compress what they give you as best as possible so as to make a profit.

Say the companies compression scheme is imperfect. They have a Huffman coding of their (imperfect) model of tokens they send, call it q(x) (that is, they think token x shows up with probability q(x)). You've determined the true distribution, p(x) (token x shows up with actual probability p(x)).

The business has tokens that show up with probability p(x) but they encode them with lg(q(x)) bits, giving an average token bit size of:

    -\sum _ x p(x) lg(q(x))

    -\sum _ x p(x) lg(p(x))

    -\sum _ x p(x) lg(p(x)) - \sum _ x p(x) lg(q(x)) = -\sum _ x p(x) lg(p(x)/q(x))

To me, this is a much more intuitive explanation. I made a blog post about it [0], if anyone cares.

[0] https://mechaelephant.com/dev/Kullback-Leibler-Divergence.ht...

To rephrase what you wrote in plain English: you are Amazon, a client uses an S3 bucket to store .zip files in them, which they pay by the byte, you re-compress and store the data as .7z files, and the KL divergence is related to zip_file_size - 7z_file_size, your "win".