For the binomial it is just the central limit theorem.

The number of successes in N trials is the sum of N iid RV taking the value 1 with probability p and 0 with probability (1-p). These have mean p and variance p(1-p). Hence the distribution of the sum tends to normality with mean Np and variance Np(1-p).

(the hypergeometric case is dealt with by first using the binomial approximation when appropriate that by the normal)

CB