How can ingesting a prion "infect" someone?
Prions are misfolded proteins with abnormal tertiary or quaternary structures. That grants them resistance (to some extent, at least) to proteases (1).

Also researchers believe that prions are able to replicate (2), by changing the structure of other proteins.

Regarding the gastrointestinal barrier, that isn't exactly true. It has been shown that small quantities of intact proteins do cross the gastrointestinal tract in animals and adult humans (3), and that this is a physiologically normal process required for antigen sampling by subepithelial immune tissue in the gut



So, the resistance to the proteases and the ability to replicate in certain conditions might explain the odds of a prion crossing the gastrointestinal tract and infecting an individual.



References:


http://www.ncbi.nlm.nih.gov/pubmed/24338008
http://www.nejm.org/doi/full/10.1056/NEJM199609193351218
http://www.ncbi.nlm.nih.gov/pubmed/3060169