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?...
The intuition lies in realizing that the flow itself distorts the length of tangent vectors as it moves, but not points because points have zero lengths.
So $D_Xf = Xf$, the directional derivative, where $f$ is a function. This is because you can directly compare the different values of $f$ as the flow moves a point. But you can’t directly compare tangent vectors as the flow moves a point. A given tangent vector at a point will be distorted by the flow itself as the flow moves the point. This distortion is represented by the Jacobean of the inverse flow in the pullback. (The Jacobean is nothing but change of variables from the inverse flow map perspective.)
You have to account for this distortion when computing the X-directional derivative of Y. This is what the second term in $D_XY = XY - YX$ does. The negative sign arises because you’re looking at the Jacobean of the inverse flow close to the identity matrix. The first term XY assumes that the only source of distortion in Y is due to the vector field, not the flow.
For example, consider a flow that shifts an Euclidean space at constant velocity isometrically. X is a constant so $YX = 0$ and we’re back to the directional derivative XY.
See the proof on p.388 of Loomis and Sternberg. You can interpret XY as X-directional derivative of Y, a vector of functions, or as a composition of derivatives. It’s easy to check that $[X,Y]f = (XY)f - (YX)f = X(Yf) - Y(Xf).$