Description of lists with functions in LISP
In PL theory, this is known as (a variant of) the Church-encoding of pairs.

The idea is the following: assume for the moment you only have a language with primitive  booleans (true, false, if-then-else) and primitive (first-class) functions. Pairs are not primitive, yet they can be encoded as follows.

In place of a pair $(x,y)$ we can consider the following function:
$$
p(b) = {\sf if}\ b \ {\sf then}\ x \ {\sf else}\ y
$$
It is immediate to verify that $p({\sf true}) = x$ and $p({\sf false}) = y$. So this function $p$ indeed encodes the information which lies inside a pair. Projections can then be defined as
$$
\pi_1(p) = p({\sf true}) \qquad
\pi_2(p) = p({\sf false})
$$
while the pair constructor becomes
$$
cons(x,y) = \lambda b.\ {\sf if}\ b \ {\sf then}\ x \ {\sf else}\ y
$$

Now, suppose we no longer have primitive booleans/conditional, but only have functions. We can still obtain something which behaves as a boolean through this encoding:
$$
{\sf true} = \lambda x y. x \qquad
{\sf false} = \lambda x y. y \qquad
$$
and (since now a boolean $b$ is a "binary" function)
$$
{\sf if}\ b \ {\sf then}\ x \ {\sf else}\ y = b x y
$$
It is immediate to verify that
$$
{\sf if}\ {\sf true} \ {\sf then}\ x \ {\sf else}\ y = {\sf true} x y = x
$$
and the $\sf false$ analogous.

Once you combine both encodings, you get that
$$
\begin{array}{l}
cons(x,y) \\
= \lambda b.\ {\sf if}\ b \ {\sf then}\ x \ {\sf else}\ y \\
= \lambda b. b x y \\
\end{array}
$$
and
$$
{\sf car} = \pi_1 = \lambda p. p {\sf true} = \lambda p. p (\lambda x y. x)
$$
and its $\sf cdr$ analogous.

The LISP encoding you have is very similar to this encoding: it only have a few more twists for encoding $\sf nil$ as well, but that's a minor point.

The variable name $k$ which appears there in place of the "boolean" $b$ likely stands for "continuation", which is a key notion in PL theory.