Fame is a statistic whose effect isn't always clear, and is often not listed with the other statistics on the monster's diagnostic page. When you first unlock a monster, its fame will be 0 because it has yet to make any friends or gain any fans. As time goes on and you win more tournaments, your monster's popularity will increase (along with your Breeder rank), thereby indicated by an increase in fame.
In some games, there is a chance that if you do poorly in a battle, your monster will lose fame. Some games have in-game events that hinge on your reaching a certain level of fame, while others just use it as an indication of when your stat gains will increase during training (reaching a certain fame level will do this in most games).
Furthermore, increasing in fame is how you win some of the Monster Rancher card games.