You can put a comma before "that" when the structure of the sentence is helped by the comma.
The particular rule that you have remembered is that you "don't put a comma before a relative clause starting with 'that'".
A book that I have read / a book, that I have read
This is because the pause and intonation implied by the comma would indicate that the relative clause is non-restrictive. And in non-restrictive relative clauses we tend to use "which" instead of "that".
In your sentence the word "that" doesn't introduce a relative clause. It is a subordinating conjunction. There are two such subordinate clauses. They are in parallel as objects of the verb "emphasise". To separate the parallel clauses, a comma is used. Compare this much simpler sentence with ordinary nouns, not clauses
I gave her a ball, a big red one.
There are two (indirect) objects of the verb "gave", "a ball" and "a big red one". They are parallel and the comma helps to make that parallelism clear.