We have the following two formula about the Lie derivative of a vector field:
$ \left.\frac{d}{dt}\right|_{t=0}T\varphi_{-t}\cdot Y_{\varphi_t(p)}=[X,Y]_p = (\mathcal{L}_XY)(p) $
where $\varphi=\varphi^X(t,p)$ is the flow along the vector field $X$,
and equivalently,
$ \mathcal{L}_XY=\left.\frac{d}{dt}\right|_{t=0}(\varphi_t^{-X})^*Y $
where $(\varphi_t^{-X})^*$ is the pull-back of $\varphi_t^{-X}$.
I have a rough idea about what this formula is saying: let $Y$ is a vector field defined along a integral curve of $X$, and we "pull-back" the vector of $Y$ at point $q=\varphi_t(p)$ to its original point $p$ and measure the change rate w.r.t $t$.
But such explanation is quite forced and can not satisfy me. Can anyone provide a intuitive explanation of the Lie derivative?...