The following is extracted from the book 'Lipschitz Algebras' by N. Weaver. (page $36$)
It is standard that a bounded linear functional $\phi \in Lip_0(X)^*$ belongs to the predual of $Lip_0(X)$ if and only if it is continuous with respect to the weak$^*$ topology.
How to prove the above statement?
Note that $X$ is a pointed metric space (We denote this point as $0$). $Lip_0(X)$ is the set of all real-valued Lipschitz functions which satisfy $f(0)=0$.
This is a general fact. Let $Y$ be any Banach space, $Y^*$ its dual (this is ${\rm Lip}_0(X)$ in your question) and $j_Y \colon Y \to Y^{**}$ the canonical inclusion map. Suppose $y^{**} \colon Y^{*} \to \mathbf K$ is a linear, weak$^*$ continuous functional (note that $y^{**} \in Y^{**}$ as weak$^*$ continuity implies norm continuity). We will show that $y^{**} \in j_Y[Y]$. As, $y^{**}$ is weak$^*$ continuous, there are $y_1, \ldots, y_k \in Y$ and $C> 0$, such that $$ \def\abs#1{\left|#1\right|}\abs{y^{**}(y^*)} \le C \max_i \abs{y^*(y_i)} $$ This implies $\bigcap_i \ker j_Y(y_i) \subseteq \ker y^{**}$, hence, there are $a_i \in \mathbf K$ with $y^{**} = \sum_i a_i j_Y(y_i)$ or $$ y^{**} = j_Y\left(\sum_i a_i y_i \right) \in j_Y[Y]. $$