Overview of "Some remarks on the Lonely Runner conjecture"

Question

I'm reading this paper https://arxiv.org/pdf/1701.02048.pdf (Some remarks on the lonely runner conjecture - Terence Tao) and I think the maths is slightly out of my reach. Of chapter 3 I understand until midway through page 14, but I am still keen to understand roughly what is going on for the whole proof. So could somebody help me understand the process and techniques that Tao uses in chapter 3 (proof of the first theorem)? I understand the proof is long but is there a very broad picture that would help me understand the proof?