Besides Samelson's short 1946 research note linked by Mrc Plm, it's also useful to mention his longer 1952 survey on topology of Lie groups here (see Section 10 and references for various proofs of Weyl's theorem on finiteness of the fundamental group).
By now the whole subject has been treated in numerous textbooks and lecture notes, from a variety of viewpoints. Which approach you take depends a lot on your own background and interests. But the finiteness by itself is too limited a goal, since case-by-case study of the simple compact Lie groups computes each fundamental group in an elegant way relative to the roots and weights of a maximal torus.