This is originally due to Ljunggren and Jacobsthal, 1952, in "On the divisibility of the difference between two binomial coefficients" in Skand. Mat.-Kongr., Trondheim 1949, 42-54 (1952).
It is also attributed to G. S. Kazandzidis, who proved a somewhat stronger result. See his papers
- "On a congruence and on a practical method for finding the highest power of a prime p which divides the binomial coefficient (AB)", Bull. Soc. Math. Grèce, N. Sér. 6, No. 2, 358-360 (1965).
- "Congruences on the binomial coefficients", Bull. Soc. Math. Grèce, N. Ser. 9, No. 1, 1-12 (1968).
- "On congruences in number-theory", Bull. Soc. Math. Grèce, N. Sér. 10, No. 1, 35-40 (1969).
These proofs are elementary. The modern treatment is a bit less so, but is more illuminating. Indeed, $p$-adic analysis can lead to a proof. See Chapter 7 of Alain M. Robert's book "A course in p-adic analysis" (GTM 198, Springer, 2000). The proof given there is based on the material in the paper "The Kazandzidis supercongruences. A simple proof and an application", Rend. Semin. Mat. Univ. Padova 94, 235-243 (1995), by Alain M. Robert and Maxime Zuber.