The concept (but not the name) was introduced by Barzdin and Kolmogorov in

A. N. Kolmogorov and Y. M. Barzdin, “On the realization of networks in
three-dimensional space” in Selected Works of Kolmogorov, vol. 3, Kluwer,
Dordrecht, 1993, 194–202.

which was published in 1967. They proved that they exist via a probabilistic argument. They were then rediscovered and named expanders by Pinsker in his paper

M. S. Pinsker, "On the complexity of a concentrator'', Proceedings of the Seventh International Teletraffic Congress (Stockholm, 1973), pp. 318/1–318/4, Paper No. 318.

available here (see the appendix). He also proves they exist via a probabilistic argument. The first explicit examples were found by Margulis in his paper

G. Margulis, Explicit constructions of concentrators, Problemy Peredachi Informatsii, 9(4) (1973), pp. 71-80; Problems Inform. Transmission, 10 (1975), pp. 325-332.

and by Gabber-Galil in their paper

O. Gabber and Z. Galil, Explicit constructions of linear size superconcentrators, Proc. 20th Annual Symposium on the Foundations of Computer Science, 1979, pp. 364-370.

By the way, I learned the above history from the following lovely paper:

M. Gromov and L. Guth,
Generalizations of the Kolmogorov-Barzdin embedding estimates.
Duke Math. J. 161 (2012), no. 13, 2549–2603.