It is true that there is a lot contained in this chapter. It extends the notion of dimension to a great many realms different from those in the earlier part of the book. Please see my comments on Irene's page as part of my response to your own comments.
Perhaps we can get closer to a careful and detailed explanation of the geometry of segments and lines in today's (Friday's) class. The 45-degree line referred to for example does indeed make an angle of 45 degrees with respect to the vertical line at the point, and also with respect to a given horizontal line, namely the one perpendicular to the line we started with. The sentence you quote is certainly a mouthful, and I guess that part of the language aquisition aspect of a math course is making it possible for such to trip off the tongue, after a lot of practice.
I hope that you complete some string art and bring it to class?