(This page is intended as a standard link from individual koans, to explain intended use.)
A "koan" is a puzzle that the reader is meant to attempt to solve before continuing. This method attempts to reflect the research which shows that you're much more likely to remember a fact or solution if you try to solve the problem yourself before reading the solution. Succeed or fail, the important thing is to have 'tried first.
Trying to solve koans is particularly important on LW (and some other places), which is that since conclusions about rationality often sound obvious in retrospect, it's hard for people to visualize the diff between "what I believed before" and "what I believed afterward". Explicitly having some idea of this difference is important to learning.
To use the koan as intended, try to say your own answer to the koan - ideally whispering it to yourself, or moving your lips as you pretend to say it, so as to make sure it's fully explicit and available for memory - before continuing. Afterward, try to consciously note the difference between your reply and the post's reply, including any extra details present or missing, without trying to minimize or maximize the difference. Again, the key idea behind a koan isn't to get it right, or even necessarily agree with the poster's own answer afterward - the key idea is to notice the before-and-after difference, instead of letting it blur into hindsight.