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 $15 and the breakeven rank is #5, that means that if the player finishes in 5th place, they'll pay out $15.  If the player finishes in better than 5th, they'll pay out higher than $15, and if they finish lower than 5th, they'll pay out less than $15.  Therefore, you should purchase shares of a player at the price of $15 if you think they'll rank better than 5th