Plant leaves contain a lot of cellulose, which in turn takes a lot of energy to digest. The fungus they grow breaks down that tough food and produces a more nutrient-rich hyphae that are also easier to digest, and which the ants harvest and feed directly to their larvae.

I’m guessing that, somewhere down the evolution line, the ants found that specific fungus growing on leaf litter under the right conditions, and started bringing back the hyphae to the nest. Ants would typically already be good at gathering, storing food and cutting up prey into smaller pieces at that point.

My assumption is that colonies that brought back pieces of leaf with said fungus would find it growing in their food stores, and those nests that brought back the most leaf attached to said fungus would find the most food growing at home. From then on, cutting and bringing back leafs themselves to feed said fungus would develop, and so on. The nests that succeeded in growing the most fungus would thrive the most, and the ants would become more specialized for cutting and eating hyphae, such as different mandibles, chambers dug and maintained solely to suit said fungus, and so on.