How does the grid store energy?

Strictly speaking, the electricity grid itself stores very little energy: electricity has to be generated at almost the same moment it's used, because supply and demand on the grid must stay balanced to keep the frequency stable.
What we mean by "grid storage" is a network of separate storage technologies sitting alongside the grid, which soak up surplus electricity when supply is high and feed it back in when demand rises or generation dips.

What are the methods used to store energy alongside the grid?

The largest and most established method to store energy alongside the grid used around the world is pumped hydro storage. When there's excess power on the grid – often from wind or solar – it's used to pump water uphill into a reservoir. When extra power is needed, that water is released back downhill through turbines, generating electricity on demand, similar in principle to the way a Hydropower Archimedean screw generates power from flowing water. It's a mature technology and still accounts for most of the grid-scale storage capacity worldwide.

Battery Energy Storage Systems (BESS) have grown rapidly alongside pumped hydro, particularly for shorter bursts of balancing. Large-scale lithium-ion battery installations, working on the same principle as domestic and commercial battery storage systems, can charge and discharge within seconds, making them ideal for smoothing out sudden spikes or dips as renewable generation fluctuates. 

Other technologies play a role too: Compressed Air Energy Storage compresses air into underground caverns to release later through turbines; flywheel storage uses spinning mass to store energy mechanically; and thermal storage banks heat (often in molten salt) for later conversion back into electricity.
Grid-scale storage is becoming increasingly important as more renewable energy comes onto the network, since wind and solar generation don't always align neatly with when electricity is actually needed. By storing surplus power and releasing it strategically, these technologies help keep the grid stable and support the wider move toward decarbonising the energy industry.