Continuation-passing style: what is meant by "CPS'ing"?
I had only a quick look at the paper, but I believe that they are referring to moving from
$$
    t_1 \to t_2 \cdots \to t_n
$$
to
$$
    t_1 \to t_2 \cdots \to \lnot \lnot t_n
$$
where $\lnot t = (t \to r)$ for some fixed return type $r$. In the paper, they choose $r = \sf Type$. So, we get
$$
    t_1 \to t_2 \cdots \to (t_n \to {\sf Type}) \to {\sf Type}
$$
In the case of the state monad we get
$$
    state \to ((a \times state) \to {\sf Type}) \to {\sf Type}
$$
which is isomorphic to (exchanging the order of the arguments)
$$
    ((a \times state) \to {\sf Type}) \to state \to {\sf Type}
$$

Anyway, Sect. 2.3 of the paper shows the actual transform.