Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem

I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure Theory

I was confused by the highlighted part. I hadn't learned the Banach-Alaoglu theorem before, and I found the statement below

$X$ is Banach space. Any closed unit ball in $X^{\star}$ is compact w.r.t. the weak topology.

I'm not sure if it's the right one. If it is, then the Banach space must be $C(K_{j})$. What is the closed unit ball in this situation? and why does the weak compactness in $C(K_{j})^{\star}$ imply the existence of convergent subsequences? Any help will be appreciated.