You can add hay to a silo by walking up to it with a stack of hay.
To remove hay from the silo, you need to use the hopper in a coop or barn when the feeding trough is not full. Stand next to the hopper with a tool in your hand, or you'll just put the hay you took out back into the silo. If you have auto-fill (upgraded to deluxe), you can remove some hay from the feeding trough with a bomb.
When I have a lot of extra grass and my silo is already full, I remove the hay and put it into a chest, then cut more grass to refill the silo. The chest can store much more hay than a silo.
It would be nice if it were easier to cut and store hay. Unfortunately, most people do not focus on raising animals, so it is not a priority for the developer.