these rat quests unlock sewer gates as fast travel points throughout the map. there are 6 gates in total.
to solve the puzzle, simply look around the quest area for a highlighted cinderblock [as pictured]. interact with it to place the items.

• Isenkram's hardware store: cheese - place it on the white feeding dish
  
• Karl's farm: oatmeal, apple, fish sticks
  
• Harbor: pink dollhouse obtained from Solruna's antique store
  
• Forest: energy drink - give it to Dora the sheep
  
• Rats on the highway: toxic barrel obtained from Barbara's third trade deal

These quests do not spawn one after another! Progress further through the campaign so they become available to you gradually.