Article is a bit light on the “how” part. Found this over at https://www.wired.com/story/the-rise-of-dna-data-storage/ :

“What Catalog does, instead, is cheaply generate large quantities of just a few different DNA molecules, none longer than 30 base pairs. Then it uses billions of enzymatic reactions to encode information into the recombination patterns of those prefab bits of DNA. Instead of mapping one bit to one base pair, bits are arranged in multidimensional matrices, and sets of molecules represent their locations in each matrix.”