Why does 0! = 1 ?

It's because $$(n-1)!=\frac{n!}{n}$$

That means :

$$\\6!=1\times 2\times 3\times 4\times 5\times 6=720 \\(6-1)!=5!=\frac{6!}{6}=120 \\(5-1)!=4!=\frac{5!}{5}=24 \\(4-1)!=3!=\frac{4!}{4}=6 \\(3-1)!=2!=\frac{3!}{3}=2 \\(2-1)!=1!=\frac{2!}{2}=1 \\\textcolor[rgb]{0,0,1}{(1-1)!=0!=\frac{1!}{1}=1}$$