Division of proteins
This is not a very helpful passage, I agree.  Some proteins are cleaved by other proteins (a protein that cuts another protein is called a protease).  Other proteins cleave themselves.  Otherwise, proteins are quite stable and don't just fragment wily-nily.

In the case of the first group (proteases acting on other proteins) the coagulation cascade may be a good place to start for you.  Precursor proteins in this pathway are converted to active enzymes by a chain reaction of protein 'division' and other protein modification.  This page has other interesting cases of this process, such as in food digestion, which may be what you're looking for.

Some of the second group are self-cleaving such that an intervening sequence is removed, and the ends are pasted back together.  (This is analogous to how introns are spliced from messenger RNA and the exons are spliced together.)  Such intervening protein sequences are called 'inteins'; the removal of one intein can generate one functional protein, so this isn't protein division.    However, some viruses encode self-cleaving regions that will split the mother protein into two or more pieces.  A famous example of this process, now used widely in biotechnology, is the 2A peptide, which is used to 'split' a single transgenenically-expressed protein into 2 (For instance if you want to express a protein and a marker such as GFP simultaneously, you could employ the 2A peptide to link these together.)

The first group may be of more interest to you because of the known role of these proteases in human health.  Also, in your question it sounds like you also may be referring to random 'fragmentation' of proteins, which is not currently known to be an important contributor to protein breakdown in the human body, even in the stomach.  Enzymes such as pepsin are responsible for digesting proteins.