The setting is that we have a pair of vecor fields $X$ and $Y$ on a manifold $M^n$, and $\phi(t) = \phi_t$ is the local flow generated by $X$. The orbit $\phi_t x$ of $x$ is the integral curve of $X$ that starts at $t=0$ at $x$.
The Lie derivative is then defined as the limit
$$[\mathscr{L}_X Y ] = \lim_{t\to 0} \frac{Y_{\phi_tx} - \phi_{t*}Y_x}{t}$$
I do understand the intuitive explanation which says that the Lie derivative looks at the difference between the vector $Y$ $t$ seconds along the orbit of $x$ ($Y_{\phi_tx}$) and the vector $Y_x$ flowed to the point $\phi_tx$ ($\phi_{t*}Y_x$).
$\phi_{t*}$ is said to be a differential, but what exacly is this written down in coordinates? Would it just boil down to something like $\phi_{t*} = \frac{d\phi}{dt}dt$? And what the heck does this strange $*$ notation for it mean?
The asterisk subscript is often used to denote pushforward of a vector (by the differential of a map).