This result is essentially the fact that irrational rotations are ergodic. The best way to prove it is by using Fourier series.
Let f be the characteristic function of E,
Extend the domain of f from [0,1] to by making it periodic with period 1. The Fourier coefficients of f are given by . The fact that E is invariant under a translation through a tells you that for all n. But a is irrational, so can never be equal to 1 unless n=0. Therefore for all , which tells you that f is (almost everywhere equal to) a constant function. Since the constant can only be 0 or 1. Thus m(E) must be 0 or 1.