The coefficient $C(E_1, E_2, \dots, E_k, \dots, E_N, t)$ corresponds to the configuration:
$1^{st}$ particle in state $E_1$, $2^{nd}$ particle in state $E_2$, $\dots,$ $k^{th}$ particle in state $E_k$, $\dots,$ $N^{th}$ particle in state $E_N$.
Say, this corresponds to the following occupation numbers:
$n_1$ particles in state $1$, $n_2$ particles in state $2$, $\dots,$ $n_l$ particles in state $l$, $\dots,$ $n_{\infty}$ particles in state ${\infty}$.
where $n_1+n_2+\dots+n_{\infty}=N$. Then, we rewrite the coefficient for this configuration in terms of occupation numbers as $\bar{C}(n_1, n_2, \dots, n_l, \dots, n_{\infty}, t)$.
Now, the new coefficient after the action of the kinetic energy operator is $C(E_1, E_2, \dots, E_{k-1}, W, E_{k+1}, \dots, E_N, t)$, which corresponds to the configuration:
$1^{st}$ particle in state $E_1$, $2^{nd}$ particle in state $E_2$, $\dots,$ $(k-1)^{th}$ particle in state $E_{k-1}$, $k^{th}$ particle in state $W$, $(k+1)^{th}$ particle in state $E_{k+1}$, $\dots,$ $N^{th}$ particle in state $E_N$.
$i.e.,$ the $k^{th}$ particle, which was originally occupying the state $E_k$, is currently in state $W$. Therefore, the number of particles occupying the state corresponding to $E_k$ should be decreased by one, and the number of particles occupying the state corresponding to $W$ should be increased by one. (In other words, the kinetic energy operator has annihilated a particle in the state corresponding to $E_k$, and created a particle in the state corresponding to $W$.)
So, the coefficient for this new configuration in terms of occupation numbers is $\bar{C}(n_1, n_2, \dots, n_{E_k}-1, \dots, n_W+1, \dots, n_{\infty}, t)$.
