Even today a purse is not just a small bag for carrying money. It can be the prize in  a contest.

The hunter may think of his quarry as a prize. But a animal taken by the hunter may not fit into a small bag. The hunter’s purse may be the small bag he made out of the hide of his catch.

Purse also means to draw the lips together like the mouth of a purse with the strings tightened. This is what the hunter does when his game escapes.

Also, here’s the Hunter’s Purse pdf.