Popular Science has a story about food allergies, and how people often confuse allergies with sensitivities.
People with allergies accidentally produce IgE molecules that identify harmless proteins like those in peanuts, shellfish, or milk as being dangerous. That means upon ingestion, IgE are like the alarm that kicks up a massive immune response, recruiting histamines and other immune cells that kill the invader. It's this overreaction that causes your throat to close or your blood pressure to drop precipitously, or any of the other allergic symptoms that transcend one bodily organ and extend into the respiratory system or perhaps the skin or cardiovascular system. This response can absolutely be life threatening.
Not everyone with a true food allergy will have the same symptoms, but they will all be governed by IgE. Conversely, people with food intolerances, like lactose intolerance, only get digestive symptoms.