What does a Scavenging Station do and how do I use it?

You can't do anything at a scavenging station personally; it has to be operated by an NPC. From what I can surmise, you have to enter workshop mode, select a spare settler (when you have settlers, of course), and then assign that settler to work at the scavenging station. You also have to select the scavenging station, and assign it to be worked at by the settler.

Source: This question posted earlier on Stack.

Each assigned scavenging station adds one random resource to your workshop inventory for every 8 in-game hours, for a total of 3 resources per game day. It may also respawn random resources near your settlement.

Source: This post on Reddit.

The Scavenging Station gives three random junk items daily. You may have to fast travel a great distance or rest a little more than 24 hours. I tested two Scav Stations at Abernathy Farms, took everything from the Workshop's inventory, and in less than 48 hours there was Fertilizer, Melon (2), Screw, Tato, and Typwriter. One Scav Station at Tenpines Bluff gave me Cloth, Corn, Gourd, and Purified Water. One settler was tending food at each location as well.

The scavenging table will send one of your settlers out into the wasteland to search for loot. Anything they find and bring back will be put into your workbench inventory at that settlement