It is worth to mention that in his original paper, E. Warning also finds a lower bound for the number of solutions:
Satz 3. Wenn das Polynom $f(X)$ (mit $g < n$) überhaupt eine Nullstelle hat, so ist die Anzahl aller (verschiedenen) Nullstellen von $f(X)$ mindestens $q^{n - g}$.
Apart from the references provided in other answers here, an excellent survey about Chevalley–Warning theorem, its history and applications and related problems about solving polynomial equations over finite fields, written by Jean-René Joly, is accessible from
Équations et variétés algébriques sur un corps fini (L'Enseignement Mathématique (1973))
