But maybe someone better informed can explain... I've only heard James Tour ranting about it
Asymmetric synthesis was already well established before 1995. Chemists like Knowles, Noyori, Sharpless, and Evans could routinely produce highly enantioenriched molecules using chiral catalysts or auxiliaries.
What made Soai's reaction different was that it did not require an external chiral source. The chiral product catalyzes its own formation, so a tiny random imbalance can be amplified into a very large enantiomeric excess.
Other examples of abiotic asymmetry, such as Pasteur's crystallization experiments and circularly polarized light, were also known earlier.
So the Nobel press release is best understood as shorthand: Soai did not discover asymmetric synthesis, but demonstrated a remarkable form of self-amplifying, spontaneous chiral symmetry breaking in an abiotic chemical system.
When the chirality of the product is an autocatalyst to its own chirality. So if your reaction starts off with a little more left than right due to molecular randomness, the left is an autocatalyst which means there is a runaway process and you end up at the end of the reaction with a lot more left than right.
If you repeat the reaction from scratch, it is random if you end up with a left product or a right product. But the product of your reaction is not a 50-50 mix left-right! I.e. you created asymmetry in your solution from purely symmetric solvents.