There are many ways to think about this--here are a couple.
1) Rank M is the maximal number of linearly independent columns (or rows) of M. So if M is not the zero matrix, it has at least one non-zero column (or row), and a set containing one non-zero vector is linearly independent. So rank M is at least 1.
2) Rank M is the dimension of the image of M, i.e. the vector space consisting of Mv for every vector V. If even one entry of M is nonzero, then there's a vector v (try one with all zeros except a 1 in one place) such that Mv is nonzero, and therefore the dimension of the image of M is not zero.
That's the "only if" direction. The "if" direction should be straightforward.