The answer lies in our conception of magic. To most people in the modern world the first image that comes to mind is Harry Potter making things fly around the room, shooting big, violent spells everywhere. Historically, this is not how witches were seen.
Magic was almost always related to a relationship with the Devil, which made it inherently evil. The witches gained their power by worshipping Satan. By doing his bidding on Earth, he in turn granted them with extraordinary powers.
Their resulting magic was much more subtle. Most reports from Europe and the Americas allege that a certain person cast some magic upon a cow and killed it or caused some crops to fail. Magic was used to harm others, but not in the direct way that we often see in popular culture.
Take the Salem Witch Trials, perhaps the most famous example in American history. When a few girls started acting in a strange manner, screaming and writhing to draw attention, it was assumed that people had cast spells upon them to make them suffer. The results of this case aren't really important for your question, but this would be an example of a way that people believed magic took a direct and tangible effect.
Arresting and executing the witches was simply reasserting God's will on Earth. The witches were under guard and were never expected to bust out riding a broom while breathing fire. The most they could do was, in a rather lengthy time, slowly poison one's soul or cause incremental physical ailments.
So, since most of the time charges instead focused on abstract allegations of sabotage and rarely human violence, they were not too worried. Most mass-hysteria episodes coincided with difficult times economically, politically, or environmentally, but it was always easier to say, "My cow died and I hate that girl. She's a witch!" The girl could take revenge, but it'd be rather difficult for her to find the time to slowly implement her incremental magic if she's constantly under surveillance and then burned to death.
Edit: Sorry for the lack of sources and formatting, I'm a little bit new here.
For example, regarding their conception of magic he writes, "Usually the danger was seen in an organized conspiracy led by the Devil. Or the concern was witches causing harm (maleficia) through spells: raising storms, killing people or livestock, and/or causing bad luck." As people became more and more hysterical, the government almost always stepped in to counteract the Devil's influence, so it was very much an institutionalized phenomenon.
He also briefly comments on why outbreaks occurred in some places more than others. He writes, "Historians are still trying to explain the reasons for this great variety in witch hunting. Important factors could have been: the power of the central government; the independence of local authorities; tensions created by war, failing economies, or famine; and uncertainties about religious conformity."
For more info on the Salem witch trials...
It was not long before one of her Sisters, an two of her Brothers, were seized, in.Order one after another with Affects' like those that molested her. Within a fe weeks, they were all four tortured every where in a manners very grievous, that it would have broke an heart of stone t have seen their Agonies. Skilful Physicians were consulted for their Help, and particularly our worthy and prudent Friend Dr. Thomas Oakes,' who found himself so affronted by the Dist'empers of the children, that he concluded nothing but an hellish Witchcraft could be the Original of these Maladies.
As you can see they didn't believe witchcraft worked anything like we do today. Because of this their fear of a witch locked up in a jail cell was naturally much different than our's would be.
There are many more books on the Salem Witch Trials, and it really is fascinating to look at why the entire thing happened.
I'd recommend The Salem Witch Trials: A Day-By-Day Chronicle of a Community Under Siege by Maryilynne K. Roach. It's essentially a timeline of the whole thing with the historical context. It's kind of long, but very informative, and not overly academic.
For a more scholarly take try Entertaining Satan: Witchcraft and the Culture of Early New England by John Demos.
I have to ask a follow-up. Knowing how the "tests" for being a witch were tests no one could pass, for example, I have often wondered if those prosecuting witches really believed in it or was that just political rhetoric. Have we any evidence historically that the powers that be really believed in it as much as the townspeople did? Were there any prosecutors who, through their writings, let on in some way that they did not believe but were promoting an agenda of some sort?
I think it might be interesting to add something to the top answer. I'm especially addressing the period of the 16th century in northestern Italy - but many of those consideration can be extended to central Europe during the same period.
For many people, especially in rural communities, the belief in witchcraft was deeply connected with the belief in god; thus the practice of religion, the traditional rites and rituals connected with the passing of the seasons, the cultivation of the fields accordingly to the lunar phases, etc. were deeply blended together. To believe in god, the devil, angels was almost the same to those people as to believe in curses and charms. And practicing this kind of magic was not necessarily in contrast with the belief in god and the church for them.
Not only there were persons believed to have those kind of powers but they believed themselves to possess the ability to do those things, at least in certain special periods of the year. You can check this wikipedia article or rather the main source, if you have it available (unfortunately I have it only in italian), which is:
Ginzburg, Carlo (1983) [1966]. The Night Battles: Witchcraft and Agrarian Cults in the Sixteenth and Seventeenth Centuries. John and Anne Tedeschi (translators). Baltimore: Johns Hopkins Press. ISBN 978-0801843860.
Here the author mainly focuses on a group of people that called themselves "Benandanti" which I might translate with "the ones who walk in the good" and claimed to fight the evil doers. Many of them were born with a caul, which was thought of as a good omen.
In the words of one of them (I'm quoting from the wiki page, rather than translating myself from italian, but there are many other interesting testimonies from the book if you can find the engish version):
I am a benandante because I go with the others to fight four times a year, that is during the Ember Days, at night; I go invisibly in spirit and the body remains behind; we go forth in the service of Christ, and the witches of the devil; we fight each other, we with bundles of fennel and they with sorghum stalks.
Therefore you have people testifying they were walking around in spirit to fight against devils - some of them referred to themselves as "God's hounds". They admitted to perform charms and to have experienced temptation by the devil in various forms. In fact many trials in the region started with genuine interest from the church into these kind of beliefs, to determine wheter these people were lunatics or really practicing some sort of magic. Anyway those beliefs appear to have been rather common and widespread.
Those ideas survived long after the witchcraft trials ended and you can still find traces of it in local folklore. For example in many cities in northern Italy they still celebrate the new year with the burning of a straw figure representing a witch, in order to favour a good reaping season. And in italian you still say that someone is "nato con la camicia" i.e. born with the amniotic sac to signify that he is somewhat "fortunate" or even "charmed". Similar expressions survive in other central euroepan countries.
Thus people were actually living in a complex net of - sometimes hard to concilate - beliefs, taken in between the church and its desire to establish a moral and material authority and traditions rooted in the pagan folklore. Many tried to concilate those contrasting issues by making up a system of beliefs which included and mixed together a bit of everything. In this context putting witches to trial was a way to prevent famines, fight epidemics, avoid injuries and people percieved and used those trials in a way not much different than taking care of murderers or thieves, for the belief in these kind of supernatural forces was deeply rooted in their society.
I hope I didn't wander too far from the original question and this might be of interest to some.
I've been generally under the impression that often magical activities occur b/c of some form of intoxication. (I don't have any sources, and it's quite possible that I'm talking out my ass.) Is there any validity to this?
Follow-up: have there been significant periods in Western history when magic has been seen as a primarily good or helpful activity, or has it always been vilified?
For a book that may complicate your understanding of witchcraft, I recommend The Night Battles, by Carlo Ginzburg
I'm really sorry if this is already answered in the comments below, but it's late and I'd like to know: When, why and how did the common tropes of the modern idea of the witch appear? I'm mainly interested in the physical traits of a witch, including green skin, black clothing and especially the crooked nose and flying broom.
Would this question be better suited for the anthropology subreddit?
If you'd like to learn about the Salem Witch trials, with reference to the Father son duo the Mathers (who penned the two leading txts on prosecuting witches, the father to ensure we didnt prosecute innocents, the son to ensure we used empirical in addition to mystical evidence in prosecuting...the son also felt the mystical was strong evidence though and only needed additional evidence that lined up with it when prosecuting. The txt goes through 5 cases and omits the one where a preacher recites the Lord's prayer on the gallows, something a witch should not have been able to do) the New Yorker recently had a great, well researched article. im on mobile so...ah fuck it I'm getting out of bed right now.
edit: here is the article. It's by Stacy Schiff who won the pulitzer for her biography on Vera Nabokov and also was nominated for her non fiction work on Antoine de Saint-Exupéry (who penned The Little Prince). She's a solid writer and the article is well researched citing passages from sermons and detailing the trials from the accusers, the history of the people shepherding the trials from initiation through prosecutions, to the finality of the events. Really great read.
edit2: i recommend everyone who comes across this read it. i imagine a lot of people on this sub would appreciate it.