Both battery production and sourcing of the raw materials are energy intensive processes. E.g. refining lithium from either ore or brine, basically involves a long process of evaporating water, concentrating usable quantities, pumping stuff around, etc. So, you could simplify the statement above to the whole process being essentially about energy cost. It also uses water in large quantities and producing and cleaning that boils down to yet more energy intensive processes.
So, as energy gets cheaper, the whole production process gets cheaper. A lot of automotive companies are announcing plans to move their production processes to be carbon neutral. That includes using clean energy for battery production. For many manufacturers that means sourcing their own energy and turning that into a fixed cost rather than a variable cost. As it is such a big component of their overall cost, you can bet they'll work hard to lower that. And as they do, everything energy intensive they do, benefits.
Of course another way batteries can get cheaper is by improving the production process to e.g. require less rare earth materials (e.g. Kobalt), using less water, etc. That too is going to happen. But bringing energy cost down is already big deal. Long term alternative battery chemistry, improved production processes, cheaper energy, all work together to bring cost down further. The question is not if but when. The key metrics here are simply $/kwh. That used to be quite high; it's now around 100$. What happens when it hits 50$ or 25$? How long will it take to get there? These are interesting questions considering that the battery is by far the most expensive component in an electrical car. Also interesting for ICE car manufacturers still hoping to keep on producing ICE cars for another decade or so. Having a lot of cheap EVs to compete with is not going to be good for them.