The second highlighted line makes sense only when you think of the goal, which is how you have to do induction proofs. It's not something you'd think of going sequentially through the process, but a leap to the goal.
We have 2*k! + 2, and want to get to (k+1)! (that is, the former has to be less than the latter). How can we link them? Well, expressing the latter in terms of k!, it is (k + 1)k!, so that's the expression we need at this point; we just have to justify the claim that 2*k! + 2 < (k + 1)k!.
So the hard part is actually the "reason" part (and filling the gap as we read, to see that that is actually true). Can you see why k >= 5 implies 2*k! + 2 < (k + 1)k! ? Give it a try. (I personally wish they had written at least one more step in there.)