Fermat's little theorem. I'm not aware of an intuitive explanation unfortunately.
I think the author probably should give the definition of the function first, and then say it fits the equation "x^λ(n) ≡ 1 (mod n) for any x such that gcd(x, n) = 1" due to the theorem, insteads of the other way around (λ(n) is a number that match this equation).
(Also, it probably is more intuitive to say "x^λ(n) - 1 is a multiple of n" than "x^λ(n) ≡ 1 (mod n)".)