As has been gone into, the group p-1 under multiplication modulo p is such that every element has an inverse. However, in the equation X^2 = 1 Modulo P, there is only two solutions, X = +1, and X=-1, SO THAT THESE TWO ELEMENTS ARE THEIR OWN INVERSE! 1 we can forget about, but -1 is its own inverse, and is present only once! Thus its inverse is not there to cancel it out, and (P-1)! = -1, Modulo P.
PS: Wilson is not thought to have proved this theorem. Rather he was the first to notice it and show it to his professor, (Dr. Waring?). Wilson went on to become a lawyer, which, no doubt, in those days as in those of Gallieo, was a much more lucrative profession than mathematics! (As Gallieo's father thought.)
PPS: Mathematics is rather interesting in that Wilson gained immortal fame with a name theorem that he never proved! But in math it was never thought that these things are named in a logical way! Fermat never offered the world a proof of his "Last Theorem." As for imaginary numbers, I knew a student in engineering, who's professor told him, "This is a very unfortunate name, imaginary numbers are as real as real numbers." (But then again, why would the opinion of a mere engineer affect anyone in the math department?)