There's an expository paper by Harald Helfgott that goes into earlier work, more foundational stuff, etc — even has exercises: https://arxiv.org/abs/1710.04574

(Originally presented for the Bourbaki Seminar, so not exactly for the non-mathematician, but has more context than reading the paper directly.)

Those are just the corrections, not the full paper.

This is the full paper the OP was referring to: http://people.cs.uchicago.edu/~laci/17groups/version2.1.pdf

The fixes linked above are mentioned in the WARNING below the abstract and explained at 18.1.

On page 2, in the "coherent configuration" section, what does the x with the - through it mean? I haven't been able to search my way to an answer