When you distill something, you are boiling it down to its essence — its most important part. Whether it's alcohol or ideas, the distilled part is the most powerful.
The original meaning of distill comes from the process of making alcohol, known as distilling, in which all the impurities of a substance are vaporized and its pure, high-alcohol condensation collected. Distill eventually came to mean any process in which the essence of something is revealed. If you take notes at a lecture and then turn them into an essay for your professor, you're distilling your notes into something more pure and exact. At least, that's what you hope you're doing.