It is worth to mention that in his original paper, E. Warning also finds a lower bound for the number of solutions:

![enter image description here][1]

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))][2] 


  [1]: https://i.sstatic.net/XP40k.jpg
  [2]: http://www.e-periodica.ch/cntmng?pid=ens-001:1973:19::6