Pest in food establishment can cause foodborne illness because

Pests in food establishments are usually very good vectors of many types of diseases, not just food borne illnesses (rats for example used to carry the fleas that carry the bubonic plague). Pests are very good at contaminating foods, as they will not only contact food in storage, but they will eat (so contact with saliva), lay eggs or have their young around, and will defecate and urinate. This means that it is highly likely to cause some food borne or other illness.