That's not such a rigorous answer:

Imagine you have a random sample with n observations x_1, ..., x_n, independently and identically distributed according to some distribution with mean mu and variance s^2.

The sample mean is sum(x_i)/n (the expected value is mu as one would hope). Doing some manipulations we find that this has variance s^2/n, i.e. a large n means a small variance, so larger samples are more tightly clustered around mu.