The best explanations for the professed purposes of the court can be found in Act Three, as Danforth tries to explain it to the people assembled there. He declares that the purpose of the court is to uphold the law, as stated in the Bible. The Puritans live in a theocracy, where their religious laws are enforced by the courts; hence, if you break a moral or religious law, you are punished in court, in jail, or through other societal means. Danforth means to, as he says, enforce the law "as written in the holy Bible."
Now, in the case of witchcraft, Danforth explains very clearly why there are no witnesses that are needed. He says that in the case of witchery, there are only two witnesses--"the witch and the victim." The witch herself is certainly not going to come forth and testify against herself, so the only other person that we can rely on to testify of what happened is the victim. And, as Danforth states, "they do testify, the children certainly do testify." Sometimes they testify that the witch's spirit comes into their bedrooms at night and does things to them there--in that case, only the victim can say what happened. So, whereas in a normal situation, witnesses would be called to describe the crime that occurred, no witnesses ever really see the witchcraft, except for the victim.
I hope that those thoughts helped to clarify things for you; good luck!