According to CBC:


  Mad cow disease is the common name for a condition known technically as bovine spongiform encephalopathy, or BSE. [...] The only known source of mad cow disease is from animal-based feed contaminated with tissue from a diseased animal. The original source of BSE is believed to have been feed containing tainted meat from sheep with a related disease called scrapie.


As prions are proteins, I would expect any that were contaminating food to be broken down to amino acids by the proteases in the digestive system. Why does this not occur? 

Proteases are enzymes in your digestive system that help break down food, acting like molecular-sized scissors that cut up proteins.
Proteases have clefts, or subpockets, into which proteins fit, where the substrate (protein) gets cut.
Infectious or pathogenic prions are resistant to proteases, because of their three-dimensional shape, which hides away parts of the prion that would normally fit in proteases and which would cause the prion to be digested.
Prions that do not cause disease — normal prions — have a different three-dimensional shape that allow them to fit into proteases, and so they are not resistant to digestion:

A wealth of evidence contends that the infectious pathogen causing the prion diseases, also referred to as spongiform encephalopathies, is solely comprised of PrPSc, the pathogenic isoform of the prion protein (21-23). Both PrPSc and its normal cellular counterpart, PrPC, are encoded by a cellular gene (2, 19). Physical and molecular characterization of PrPSc and PrPC has failed to reveal any chemical differences between the two isoforms (32). However, PrPSc acquires distinctive conformational characteristics upon its conversion from PrPC. Whereas PrPC is soluble in most detergents and can be easily digested by proteases, PrPScis insoluble in detergents and maintains a protease-resistant core, designated PrP27-30, which polymerizes into amyloid (25).

Dr. Neena Singh also discovered that prions "piggyback" or attach to another protein called ferritin, as they make their way through the digestive system:

Disease-causing prions are thought to have passed into people when they ate beef from infected cattle, triggering the brain wasting condition called new-variant Creutzfeldt-Jakob disease, or vCJD. But researchers have not been sure exactly how prions enter the body.
To find out, Neena Singh and her team at Case Western Reserve University in Cleveland, Ohio, mimicked the process of eating and digesting infected meat.
They mashed up brain tissue that contained prions from patients who had a form of Creutzfeldt-Jakob disease. They then exposed it to a range of harsh digestive enzymes from the mouth, stomach and intestine, which normally break proteins into pieces.
Prions, which are known to be enormously tough, escape this attack almost unscathed, they showed, as does a second type of protein called ferritin, which stores iron and is abundant in meat. The two proteins seem to stick together, they report in the Journal of Neuroscience.
The researchers next added the digested slurry to a lab model of the human gut: a growing sheet of cells from the intestinal lining. By attaching fluorescent tags to the two proteins, they showed that they are transported through the cells hand-in-hand. "Prions probably ride piggyback" through the gut wall into the body, Singh says.

Attaching to ferritin may provide additional protection from digestion, insofar as this removes prions from the digestive system, where proteases are concentrated.
