One of the most important aspects of Shakespeare's plays is his ability to capture not just popular beliefs of the time, but his gift of imbuing these beliefs into his dramatic presentations. We see how the Elizabethan audience perceived the supernatural, how Shakespeare "pandered" to these beliefs, and how he also changed the way his audience saw some of the supernatural creatures they were certain existed, with a power to affect their lives.
...during the age of the living Shakespeare, the Renaissance... practically every type of written word deals with the supernatural...[and] Shakespeare stands out...because of his profound contributions to literature which embody and illustrate the current beliefs of the era.
Elizabethans strongly believed in the malicious nature of witches—their service of "the evil one."
[The] belief [in witches] is the most frightening superstition believed by the Elizabethans. The belief in witches and witchcraft runs rampant during the Renaissance.
Even James I of England authored a book called Daemonologie, reflecting his deep interest in the "powers of darkness."
The belief in witches was seen in the persecution of women accused of witchcraft, and executed in Europe and even America. In Renaissance England...
Witches are...believed to be old hags who are “thought to be in league with Satan” and who possess strange powers of darkness given by the dark lord.
It was believed witches could change the weather, make themselves invisible, sicken animals, fly, and harm humans. Shakespeare influenced the beliefs of his audience while setting the mood for his stories. Witches fed a mood of evil.
...the language that Shakespeare uses to allude to witches and witchcraft is accepted as a possible fact...By using witches, Shakespeare is capable of giving evil, and the cause of evil actions, an actual physical form for the audience to focus on...
In Macbeth, Shakespeare uses the witches to trick our tragic hero into believing that he can be King; they convince him, too, that he is indestructible: that no one can defeat him. The witches purposely create a false sense of security in Macbeth—as Hecate (goddess of the witches) notes—to bring about his (her) destruction.
In Act One, scene three (as Shakespeare is setting the mood), the Bard describes the nature of the "weird sisters." The witches are characterized by their actions. Because a sailor's wife would not give one of them chestnuts to eat, she puts a curse on the sailor's ship.
Rather than changing into a rat, the translations I have read (including that on eNotes' site), indicate that the witch is casting a spell. The rat without the tail seems to have some specific power; the witch says that like the rat, she will magically float in a bowl with holes in it to torment the sailor. Each "sister" thereafter promises to add something so the witch's spell is successful.
But in a sieve I'll thither sail,
And, like a rat without a tail,
I'll do, I'll do, and I'll do. (9-11)
Similar to Act One, Act Four, scene one, offers a list of ingredients the witches use to create visions to further trick Macbeth.
With the behaviors Shakespeare attributes to the witches throughout the play, it is easy to infer that the tailless rat was an "ingredient" for the spell the witch in Act One was casting—not that she changed herself into a rat. She says that "like a rat" she will carry out the spell, which denotes (with a simile) a comparison.