Comment by ndriscoll

"The meme" literally comes from a book that was offering it as an intuitive explanation of a long definition, assuming you know what a monoid is. All the laws and stuff boil down to "if you generalize the idea of a monoid a little bit, and if you have some functor+flatten+wrap forming a monoid, we call that a monad." If you don't know what that stuff means then obviously it's not for you, but if you do, then it's actually a concise way to give an intuition for "what (or why) it is," which is basically just that flatten is associative and wrap is neutral.

Like if someone says they know about rings and modules, you might say that an ideal is just an R-submodule of R, which grants an interesting perspective and gives a quick, memorable definition. But if they don't know about modules, you might not give them that definition.