In the first section of code, the author is showing some Haskell operations and showing equivalent operations in the shell. The most important ones are return, >>=, and >>. You can think of return as a kind of constructor for the monad (It also has the name "unit" and I think the cool kids are calling it "pure" these days). If you have a value, x, "return x" basically gives you x contained in the monad. >>= (also called "bind") is the "monadic operator". It just means to run a function with the contents of the monad as the parameter. The function has to return a monad containing the return value (usually by running "return", which is why it was called "return"). The monadic operator is useful for chaining functions. It basically is used when you want to run a series of functions, using the output from one as the input for the next (basically the same as the shell pipe). >> is similar to >>=, but you don't use the output from the previous function as the input to the next function. Hopefully with that explanation, you can see how it matches up.
In the next code section the author is stating the three monadic laws. They are writing the laws in Haskell and the laws in shell commands. You can ignore the Haskell unless you just want to compare it to the Wikipedia article on monads. The shell commands are the interesting bits.
After that, there's no Haskell at all. It's quite an interesting observation, but it's not nearly as complicated as it might seem.
If you say that about Milewki's talks then I don't agree. He does use Haskell notation a lot, but gives C examples as well and most of his arguments are based in category theory/general programming.
Anyway, I think that the series of talks linked in the GP are a must watch. Even for people that don't fully agree with the viewpoint of Category Theory being a "holly grail" the videos do provide a different point of understanding.
edit: Didn't see edited comment you mentioned until after I wrote that. Thanks.