Lie derivative $\mathcal{L}_XJ(Y)$ with endomorphism $J$
I guess It's the Lie derivative extended on tensors. You have to think like a Leibniz's rule and you can understand the definition.
$L_X(J(Y))$ is $X(J(Y))$, you derived the function $J(Y)$ along $X$. Next, $J(L_XY)$ is like "$Y$ derived along $X$ and $J$ not derived" and $Y$ derived along $X$ is $[Y,X]$, and so on.