The reason snake handlers handle snakes is Mark 16:17-18. They believe in the inerrancy of the bible--a phrase meaning that they believe that the bible, no matter where it came from, no matter who edited it, etc., is literally true.
17: And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues;
18: They shall take up serpents; and if they drink any deadly thing, it shall not hurt them; they shall lay hands on the sick, and they shall recover.
--More from Perspectives... by Williams. See below for link.