Some people are responding with saying it's just Shakespeare. Can I just say as a Shakespeare specialist, this is not an acceptable answer? Shakespeare did not invent these things (and it is an open question how much of our modern Macbeth was written by Shakespeare any ways EDIT: please note /u/texpeare disagreeing below).

Shakespeare was drawing on a historical tradition of how to depict witches. And I think OP is more interested (and so am I) in hearing about the origins of that, than that the specific phrasing quoted most often is Shakespeare's version.

There are a couple of spell books from medieval times. The two most famous are Picatrix WARNING BIG PDF FILE and The Sworn Book of Honorius the Mage. They don't contain eye of newt but the incantation and ritual bear some resemblance to Macbeth.

Fillet of a fenny snake, In the cauldron boil and bake; Eye of newt, and toe of frog, Wool of bat, and tongue of dog, Adder’s fork, and blind-worm’s sting, Lizard’s leg, and howlet’s wing, For a charm of powerful trouble, Like a hell-broth boil and bubble. Macbeth, 1605

The popular stereotype, at least, is attributable to Shakespeare. There's some literary debate about whether he meant some sort of plant or actual eyes of actual newts. In either case, it's not a real spell, it's just a suitably spooky poetry - totally made up by the Bard.

I've read that it all comes from the Malleus Maleficarum , published in 1487, about 100 years before Shakespeare was born. The books describes how spells are done among other things. I don't know if there is anything about eye of newt in it, but it does contain similar spells.