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.
P.S. Lucy has found the answer to the question asked about finiteness of the fundamental group (via Knapp's book), though the result is actually Weyl's theorem and not just sometimes called that. As Johannes points out, there is a full treatment in V.7 of the Springer GTM 98 by Brocker and tom Dieck which has the advantage of integrating the topological questions with structure, classification, and representation theory of arbitrary compact connected Lie groups; note their nice summary (7.13). Like other basic theorems, Weyl's theorem has been developed from a variety of directions as indicated in the answers here, though for me the actual computation of the fundamental group for each simple type is an essential part of the picture.