The simplest quartic fields are the real cyclic quartic number fields defined by the irreducible quartic polynomials $x^4-m{x}^3-6x^2+mx+1$, where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. We give an explicit lower bound for their class numbers which is much better than the previous known ones obtained by A. Lazarus. Then, using it, we determine the simplest quartic fields with ideal class groups of exponents $\le 2$.