Baiting a rat trap with cheese has become quite a stereotype due to the Tom and Jerry cartoons.
It is true that rats really do eat cheese, and it is successful as a rat bait because of its high caloric and fat content that is quite attractive to them.
But the truth is that rats will eat many different types of foods, and the bulk of their diet in the wild usually comes from various cereal grains and fruits.
But any dense, high calorie food such as cheese and peanut butter are also favorites of rats, as they are many other animals including humans.