This is what the intestine looks like when your pet swallows a fishing line.  This is called: plication of the small bowel. This small intestine has an accordion-like appearance. As you can imagine  -- you simply can't retrieve the fishing line by pulling it out through the mouth. Doing so would lacerate the small bowel, stomach, esophagus and set up a severe peritonitis, possibly leading to death.

Sooo..

With that in mind, when your pet decides to swallow a string-foreign body, please call your veterinarian and follow his/her instructions.  If your veterinarian is closed, call an emergency hospital in your area. More than likely, they will ask you to bring your pet in for immediate enterotomy (abdominal surgery).  Be prepared, this surgery can be expensive.