The liver processes them and produces the enzymes for dissolving. It can work normally until most of the liver is not functioning (5%). At that point you may get a severe burning sensation in your gut and severe diarrhea (which is your body getting rid of the fat). It used to be that medical science claimed it only happened with high alcohol consumption now they admit there is a none alcoholic version of liver disease. Oh, forgot it leads to fat deposits in the liver and all over the body internally and externally.
The liver produces bile, which breaks up fats to allow them to be digested by enzymes, some of which are also produced by the liver. A damaged liver produces less of these chemicals so fats are not digested.