On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write
\begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*}
Here, $\omega$ is the connection one form and $X,Y$ are vector fields on the principal bundle $P$.
I am just wondering how they arrive at this equation and how to interpret the quantity $X(\omega(Y))$ since $X$ is a vector field in $P$ and $\omega(Y)$ is a vector field in $\mathfrak{g}$ - the Lie algebra of the structure group $G$.
This is just the Leibniz rule for the Lie derivative - this is more clear if you rewrite it as $$L_X ( \omega (Y) ) = (L_X \omega) (Y) + \omega( L_X Y).$$
Since $\omega$ is a $\mathfrak{g}$-valued 1-form on $P$, $\omega(Y)$ is just a function $P \to \mathfrak{g}$. Since $\mathfrak{g}$ is just a vector space, we can simply take the componentwise derivative of $\omega(Y)$ in the direction $X$ to get $X(\omega(Y))$.