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.