The Sewers is a hidden area in Stardew Valley where you will find Krobus, a friendly Shadow Person who will sell some interesting items. The area is also used for changing your profession choices, as well as being important for certain quests in the game.
In order to unlock the Sewers, you will need to donate 60 items to the Museum. After donating enough items Gunter will give you the Sewer Key on the next morning when you exit your house. This key opens the entrance to the Sewers permanently.
Getting 60 out of the 95 possible items to donate can take a lot of effort, and the best way to reach this goal is by always donating everything new once you find it, instead of selling it. Always go to Clint the blacksmith to open up your geodes, since these has a high chance of containing Minerals that can be donated to the Museum.
If you see the worms in the ground, then use your till to get an item from them. This item can often be an Artifact that can be donated to the Museum.