Assume the students already know about the central limit theorem. Explain how the two theorems talk about limits in different direction: let $S_n$ be the sum of $n$ independent variables of identical distributions (real valued, with zero mean and finite variance), the central limit theorem gives a limit of the unscaled probability $P(S_n/\sqrt{n} < c)$, this limit is strictly between 0 and 1; whereas large deviation theorems give the rate of decrease of a probability like $P(S_n/n < c)$.