Fava beans can trigger a bad reaction in some people, but it has nothing to do with allergies. Instead a compound in the beans can result in a rapid breakdown of red blood cells, which ferry oxygen around the body. As such, not enough oxygen gets to the organs. This break down of red blood cells is called hemolytic anemia, and symptoms include shortness of breath, fatigue and jaundice. In very severe cases, death can result.
This all happens because some people don't have enough of a certain enzyme called glucose-6-phosphate dehydrogenase, or G6PD. That enzyme helps deal with a particular class of chemicals called reactive oxygen species, and consuming fava beans leads to a buildup of those chemicals.
Learn more about the other characteristics of G6PD deficiency from the video; it was discovered when researchers were developing a drug for malaria. There is an interesting connection between the deficiency and the disease as well, discussed in the final minute of the brief video.