Description of lists with functions in LISP

Question

I have been given the following implementation of basic list functions in LISP:

nil = (lambda (k) (k ’none ’none))
(cons x y) = (lambda (k) (k x y))
(car l) = (l (lambda (x y) x))
(cdr l) = (l (lambda (x y) y))
(null? l) = (l (lambda (x y) (= x ’none)))

As an intent to show that data can be constructed using only pure functions.

I don't understand very well this code. For starters, what's the role of k? Can you show how I can come up with this implementation? Basically, I'm looking for some intuition on how this works and if you can show in some manner how you deduce the implementation.