A mix of things.
There are storage options which fit different criteria. Some are very responsive to changes in demand (supercapacitors, inertial flywheel storage), but have high costs, bad wear characteristics, and low total capacity.
Some are responsive and immensely efficient (high energy-in:energy-out ratios), such as pumped hydro, but suffer from siting and environmental restrictions.
Some are slow and inefficient but scale wonderfully. Thermal energy storage, either for direct heating or for generation, is a case in point. District thermal energy storage has been used for residential & commercial space. Molten salt thermal storage can bank heat for weeks, but suffers Carnot efficiency losses of about 70%. It's cheap, simple, and non-toxic (though will sting if it gets out).
Compressed air energy storage is a lot like pumped-hydro, though it might possibly scale better and be suited to use in exhausted natural gas reservoirs. Here a problem is physics and Boyle's Law: compressed gasses heat, expanding gasses cool. This means either that you're losing a lot of energy to the environent and freezing your regeneration equipment, having to deal with high temperatures in storage (the possibility of triggering explosions in spent-gas reservoirs might be a risk), or have to re-heat expanding gas to prevent freezing in turbines.
Fuel synthesis -- hydrogen, natural gas, or liquid hydrocarbon analogues -- is an option which provides very-long-term storage capacity (proven to 100s of millions of years), but is inefficient (about 15% round-trip), and has proved uneconomic or impractable to date, despite over 50 years of research. I hold out some hopes for this.
Load shifting, and moving to a model of dispatchable demand rather than dispatchable supply might also apply.
Battery storage is another option, and will likely see some use. It's expensive, is limited by chemistry, and most of the options suffer from considerable side effects or risks. Some of the more interesting alternatives I've seen, liquid metal and molten salt batteries, both reasonably well-suited for large, static, grid-scale infrasturcture, have proved highly problematic. See Donald Sadoway's start-up, Ambri, pursuing a liqud-metal design touted loudly a few years earlier: https://www.greentechmedia.com/articles/read/ambri-is-still-...
LiON is proven, but its light weight makes it favourable for mobile and transport applications. There's a limited amount of lithium that's readily available, and reserving that preferentially for where it's best suited is probably wise.
Among the problematic aspects of batteries: electrolyte scarcity, toxicity (lead-acid, etc.), high-temperature designs (liquid metal / molten salt), reactivity, fire/explosion hazards, and fundamental energy-density limitations, are all constraints.
My read is that all sides of the power equation will change markedly over coming decades: supply, demand, transmission, storage. We've been spoilt in ways we don't yet appreciate.