Cov(X,Y) = E((X - E(X)) * (Y - E(Y))) ?
This is intuitive. Let us suppose that they have positive covariance. That means we expect that an increase in X is accompanied by and increase in Y. So, if x>E(X) there is a tendency to have y>E(Y) thus making the covariance positive.
Also, if x<E(X) we expect y<E(Y), and again Cov(X,Y)>0.