The question is related to setting up a recurrence to model "How many slices of pizza do you get afterncuts"?

I'm having trouble following their reasoning. I've annotated the picture in question with the parts I don't get:

Could someone explain the underlined points?

Why is: splits $\displaystyle k$ old regions $\displaystyle \Leftrightarrow$ intersects old $\displaystyle n-1$ lines at $\displaystyle k-1$ different places, and so on?