(n+1)! is equal to (n+1)*n*(n-1)*(n-2)...*3*2*1. Basically, it multiplies together every positive integer that is at most n+1. This is the definition of the !, or the factorial function.
(n+2)! is equal to (n+2)*(n+1)*n*(n-1)*(n-2)...*3*2*1, using the same definition. This thus is equal to (n+1)!*(n+2), so we have:
(n+1)!/(n+1)!(n+2), or:
1/(n+2)
This is our final simplification.