Compactness of the closed unit ball of ${\rm Lip}_0(X)$ in the topology of pointwise convergence implies that ${\rm Lip}_0(X)$ is a dual space It is proven that a closed unit ball in the set of real-valued Lipschitz functions ${\rm Lip}_0(X)$ defined on a Banach space $X$ is compact for the topology of pointwise convergence. 
However, I fail to understand that this implies that ${\rm Lip}_0(X)$ is a dual space. In other words, the space ${\rm Lip}_0(X)$ is a predual. 
Can anyone explain to me?

This follows from a general fact that if $B$ is a Banach space and $E\subset B^*$ is a set of funtionals that separate points in $E$ and the closed unit ball of $B$ is compact with respect to the weak topology introduced by $E$, then $B$ is isometric to $({\rm span}\, E)^*$. See this paper.
In your case $E=\{\delta_x\colon x\in X\}$.
