The phrase "to win this beauty" means to be victorious or successful in obtaining something that is considered attractive or excellent. Full definition
The car won't win any beauty contests but it starts right up and has good tires. (spreadmyad.com)
The PS4 port of the game won't win any beauty contests, but it ran smoothly and put everyone on an equal playing field when it came to remaining connected. (mmogames.com)
As a player you will never win a beauty contest in such position. (justarsenal.com)