How to find the Branch factor of 8 Puzzle
The relevance of the blank space is that a move is encoded by the transition of the blank space.  So for example, if the blank space is in the center we have 4 possible moves:





This is what the author is talking about when he says 


  "2 from each corner for a total of 8 corner moves."


The blank space is what's in the corner. Here's one of the 4 possible corner positions: 



As you can see, when the blank space is in the corner there are two possible transitions.

Continuing,


  "3 from center of each side for a total of 12"


There are 4 centers of a side and each has 3 possible transitions.



I hope this combined with Jeffrey's weighted average explanation answers the question.