Breakeven rank is the place that a player will have to finish in order to pay out more than the price you're paying for the player.

For example, if a player's current share price is $13 and the breakeven rank is #7, that means that if the player finishes in 7th place, they'll pay out $13.  If the player finishes in better than 7th, they'll pay out higher than $13, and if they finish lower than 7th, they'll pay out less than $13.  Therefore, you should purchase shares of a player at the price of $13 if you think they'll rank anything better than 7th.