Unsurprisingly, this is a pretty complex topic, and one where the view of historians has changed over the last few decades. The "popular" view often attributes witch trials to mass panics or delusions, but that really isn't supported in much of the evidence. The advantage of studying witchcraft in this time period is the wealth of evidence about the process of accusation and trial. In England, where the majority of our sources originate, pamphlets about significant cases of witchcraft were printed and sold to the general population. Many of these pamphlets survive, and provide a key source for examining how people thought about witchcraft.

The first step to examining these sources is to drop any sort of modern opinion about the validity of witchcraft; in particular, be careful not to think of believers of witchcraft as uninformed medieval peasants. Instead, consider that European witchcraft was, at its peak (1400-1600s), a social reality. Your average Englishman would have heard stories about witches, read pamphlets about them, and maybe even known someone who had been involved with a witch (either as enchanter or enchantress). So, when a person believed themselves to be bewitched, they did so earnestly and would attribute their suffering to the actions of some witch. Negative applications of witchcraft, termed maleficium, could manifest itself as sickness, death, poor harvests, accidents, infertility, or any number of broad problems.

So how did your average European go from believing themselves to be bewitched to accusing someone of being a witch? In this time period, small communities often relied on each other for practical issues: borrowing food or tools, taking care of each other when sick, and helping with birth. In almost every case we have, the person accused of being a witch was known to the accuser. Many times the accused would already have a negative reputation (making them more vulnerable to accusations; some women had been repeatedly accused of witchcraft for years before being brought to trial). Many accused had grudges or recent disagreements with their accuser. This may seem irrational to the modern observer, but accepting witchcraft as a social reality makes it seem natural. As an example of this interaction, lets look at St. Osyth, one of the most prominent examples of witchcraft we have.

Ursula Kemp was accused of being a witch by several of her neighbors, in the village of St. Osyth in 1582 (a century before your time period, but still relevant). Kemp already had a reputation: at one point, she was believed to have magically cured a neighbor's baby who was very ill. She was formally accused of maleficium after a neighbor's newborn fell from his crib and died. The child died very shortly after Kemp argued with the mother about how Kemp was not selected to wetnurse the child (a sign of trust and respect). The mother, hysteric about her child's death and knowing of Kemp's reputation, immediately connected the timing of her argument with the child's death. After a lengthy trial period and several confessions, Kemp was hung (along with several other witches who were also dragged into the trial).

There are examples of accusing witches simply to get back at them, but this was a risky practice. If the accuser was discovered as falsifying their claims, they were at great risk of prosecution. There are several notable cases of "false" possession where afflicted children revealed they had faked their illness, and they and their parents were mocked in a widely-distributed pamphlet. So while we cannot outright dismiss that this happened, it was far less common than the "natural" accusation of neighbors. The only case where this happened somewhat frequently is in witches' confessions: prosecutors would pressure witches to accuse members of their Sabbat and reveal more witches to the court. Often, women who knew they were facing death would "confess" that certain women were also witches. In a few cases, this included accusing the very person who accused them!

Hopefully this answers your question, and let me know if you want a few more specific examples.

Sources:

Barbara Rosen, "Witchcraft in England: 1558-1618," University of Massachusetts Press (1991).

Robin Briggs, "Witches and Neighbors: The Social and Cultural Context of European Witchcraft," Penguin Books (1998).