In microarray normalization, why is the normalization factor this?
$G_k^{'}$ and $R_k^{'}$ are normalized values of $G_k$ and $R_k$. 

Take say G as $[1,2,3,4]$ and R as $[100,150,200,400]$ as your values and you want to normalize them. This is scaling one of them onto the other and bringing them on an equal level to compare. So in your case the factor is $85$ units. So a unit of R amounts to $85$ units in G. 

So to scale G to R, multiply G by $85$ or you can scale R to the level of G by dividing R by $85$. So the values are either $[85,170,..]$ and $[100,150,200,400]$ or $[1,2,3,4]$ and $[1.17,1.76,..]$ from our example. 

I think it should be mentioned as $T^{'} = R_i/(G_i*N_{total})$ as in log ratio from the statement in your question.