How does (k+2)(k+1) become P(k+1) in the video? 

Here is the image of the portion of the video, in which you have doubt.  

Since, P(n) = n (n +1),

Therefore, P(k + 1) = (k +1) {(k +1) + 1)} = (k +1)(k +2) = (k +2)(k +1)

Hope it helps!