Like other forms of amyloid, aggregated prions form a beta-sheet structure, where individual beta-strands are linked together by hydrogen bonds between their extended peoptide backbones, and hydrophobic interactions between the side chains on either side of the peptide backbone. The beta-strand at each end of the beta-sheet has unpaired hydrogen bonds and hydrophobic side chains on the exposed end, which act as a template to link with an incoming prion. Thus an incoming prion molecule can be unfolded as it joins with the exposed beta-strand, rather like a zip, and a new exposed edge is formed, waiting for the next prion to join.
This chain reaction is rather like normal crystalization, except that the process of aggregation also involves a conformational change from a folded globular molecule to an extended beta strand, which is catalyzed by the presence of existing beta-strands. So the aggregation of prion effectively catalyzes more aggregation in a positive feedback loop: Once a seed has formed, the aggregation process will continue to unfold and recruit normal protein molecules. Thus BSE/CJD was originally thought to be a virus as the aggregated conformation appears to be "infectious", passing its information about its aggregated state to a native folded prion molecule.
See for example:
Or just google "prion aggregation mechanism review"