I won't talk too much about what lambda functions or closures are, as you can find many good blog posts describing them in great details. To sum up, a lambda function is an anonymous PHP function that can be stored in a variable and passed as an argument to other functions or methods. A closure is a lambda function that is aware of its surrounding context.

He includes several examples including how they would work with a few of the array functions, an implementation of the Y-combinator method (as written by Stanislav Malyshev) and how they can be used to create dependency injection functionality.

One of the ways to do it is to use Y combinator function, which allows, by application of dark magic and friendly spirits from other dimensions, to convert non-recursive code to recursive code. [...] Doing Y-combinator in PHP was attempted before (and here), but now I think it works better. It could be even nicer if PHP syntax allowed chaining function invocations - ($foo($bar))($baz) - but for now it doesn't.

His (less than ideal) first method throws in some variable variables and a separate factoral function and his second passes in a factoral value of itself to call itself the correct number of times.