Interestingly enough, in the real events in Salem it was a medical doctor who made the diagnosis of witchcraft, not the Puritan clergy. The clergy in Massachusetts were opposed to the trials throughout, it was those who did not understand what the Bible actually said who jumped to the conclusion that witchcraft was involved. Witchcraft was a common pseudo-religion of the time in both Europe and the colonies, and legally was heresy in most of Europe but a civil crime in Britain and her colonies, which is why the civil courts were able to conduct the trials. Witchcraft was assumed to consist of three degrees, the first being charms, superstitious rhymes and other harmless practices which were not illegal, the second being "black magic" or an appeal to the devil to aid one in harming another. This was also termed "malefic magic", or a spell intended to harm someone (usually involving a doll or puppet). That and the third type, the alleged pact with the devil, were civil crimes in British territories.
A good resource to answer this question is looking at the prologue to the play; Arthur Miller gives a great commentary to why Salemites were so prone to believe in witchcraft, and to use it as a tool for blaming others. So, take a look at those explanations that come before Act One even starts.
A couple things that Miller mentions relate to the Puritan religion that these people practiced. They believed in the bible, literally, and took its words to heart. And, the devil is a very present force in the bible; hence, they believed not only in God, but in the devil also. Satan himself spoke with and tempted Jesus in the New Testament, and there are multiple instances of people being "possessed" by Satan's spirits. So, because they believed in these things, witchcraft was a completely real possibility for them. Witchcraft had been found in their communities before, and was feared among all.
Another thing that Miller mentions relates to how people dealt with guilt. He states that in their religion there wasn't a real good way to express sins and not be condemned for them. So, the accusations of witchcraft allowed people to
"express publicly his guilt and sins, under the cover of accusations against the victims."
For example, Abigail admits that she had been dancing and casting spells, but, it wasn't her fault, because a witch had made her do it. If anyone else had done something wrong, they could confess it, and get the guilt off of their chests, and blame someone else for bewitching them. It takes the blame away from them.
One last reason you might want to consider is the huge amount of contention that was evidenced in Salem at the time. Miller goes to great lengths to reveal all of the bickering, fighting, and underlying hatred that existed amongst the townsfolk. People were angry over land, over elections, over wills, over salaries and sermons, and upset over children dying, jealous over someone else's children or land, etc. All of these issues bred underlying bitterness between people; the accusations provided an outlet for them to express those frustrations. Take for example Putnam prompting his daughter to cry out against Jacobs, which, as he states, "gave him a fair piece of land." If Jacobs is accused, Putnam can buy up his land. Rebecca Nurse is arrested because Mrs. Putnam is jealous of all of her children and resents her for it. Elizabeth is arrested because Abby hates her. So, witchcraft is just a way to stick it to someone that you don't like.
I hope that those thoughts helped a bit; good luck!