Just by definition of the factorial,
n!= (n-1)! \*n
If you want to find a common denominator of
a/n! + b/(n-1)!,
you can use this to see that n! is their common denominator and
a/n! + b/(n-1)! = a/((n-1)!\*n) + (b\*n)/((n-1)! \*n) = (a+bn)/ n!
In your reference, this trick happens twice, once with n! and (n-1)! and the other time with (m-n)! and (m-n-1)!.