I have this book, so if there's a specific point you're not understanding let me know.
Personally I do think it makes it a little more complicated than it needs to be, talking about the "major scale of the lower note" (p.22).
You don't need to refer to a scale to understand intervals - except you do need to know the natural notes (C major scale) and that there's half-steps between B-C and E-F.
Here's a list of intervals within an octave, showing all the examples of each type in the C major scale (all ascending intervals):
NAME NOTES APART SEMITONES APART EXAMPLES
Unison 1 0 C-C
Minor 2nd 2 1 B-C, E-F
Major 2nd 2 2 A-B, C-D, D-E, F-G, G-A
Minor 3rd 3 3 A-C, B-D, D-F, E-G
Major 3rd 3 4 C-E, F-A, G-B
Perfect 4th 4 5 A-D, B-E, C-F, D-G, E-A, G-C
Augmented 4th 4 6 F-B
Diminished 5th 5 6 B-F
Perfect 5th 5 7 A-E, C-G, D-A, E-B, F-C, G-D
Minor 6th 6 8 A-F, B-G, E-C
Major 6th 6 9 C-A, D-B, F-D, G-E
Minor 7th 7 10 A-G, B-F, D-C, E-D, G-F
Major 7th 7 11 C-B, F-E
Octave 8 12 C-C
By the way, the book has one mistake in the answers (at least, my edition does).
Page 20, exercise 6: the 4th note shown on the first line is a Cb. The enharmonic equivalent of this is B natural, not B# (as the answers say)!