They didn't really explain it well on the show; but this is how I understand the dowry to work:
All of the ladies start out with $5,000. At various points in the contest, the ladies win/earn additional money for accomplishing things like getting picked first for a date ($2,000), having the challenging men pick you to make their plea ($2,000/ea)...etc. At the end of the game/show, if you get married, you win the money you've accumulated.
At least, that's my take on how it works.
.....Kelly