This is discussed in a related cstheory post:
LP relaxation of independent set
That post cites this publication:
 Nemhauser, G.L., Trotter, L.E. Vertex packings: Structural properties and algorithms. Mathematical Programming 8, 232–248 (1975). https://doi.org/10.1007/BF01580444
That publication in turn cites a few others, including these:
 Balinski, M. (1970). ON MAXIMUM MATCHING, MINIMUM COVERING AND THEIR CONNECTIONS. In KUHN H. (Ed.), Proceedings of the Princeton Symposium on Mathematical Programming (pp. 303-312). Princeton, New Jersey: Princeton University Press. https://doi.org/10.2307/j.ctt13x0wct.16
 Nemhauser, G.L., Trotter, L.E. Properties of vertex packing and independence system polyhedra. Mathematical Programming 6, 48–61 (1974). https://doi.org/10.1007/BF01580222
 Leslie Earl Trotter. 1973. Solution characteristics and algorithms for the vertex packing problem. Ph.D. Dissertation. Cornell University, USA. Order Number: AAI7406323.
You can look through these. From a quick scan of the first three it didn't seem obvious which contained a proof. If I recall, one can prove that for bipartite graphs there are optimal integer solutions (by considering the underlying max flow problem), and then, to prove the existence of a half-integral optimum for any general graph, convert it into a bipartite graph by making two copies of each node in a natural way and applying the result for bipartite graphs. Maybe there are simpler ways to go about it.
BTW, this kind of reference request, which you can answer yourself by taking some time to do a web search, is probably not an appropriate question for this forum, which is for research-level questions in TCS.