A mere claim that you don't understand something is not a valid question. You have to say precisely what in the text does not make sense to you. As for PA x PB, it's the product of the lengths of the segments PA and PB (not the cross product or anything like that).
The only flaw I find with the proof is that it does not properly introduce points C and D. The claim that is being proved is that for all points A, B, C, D on the circle, PA x PB = PC x PD.
Just to highlight an obvious point, the phrase "the power of a point with respect to the circle" is an indivisible term that is defined at the top of the page. The meaning of this phrase comes from that definition, not from the words that make up the phrase. You are not supposed to figure out what it is by trying to find how power, as defined in the dictionary, fits into geometry. The concept can as well be given the name "mimsy borogove".