"To earn that privilege" means that you have to do something deserving or prove yourself in order to be given a special advantage or benefit. Full definition
When I was coaching, my student - athletes were well aware that less than 2 percent of that population earn the privilege of playing a college sport. (entrepreneur.com)
At least three prisoners at risk of suicide were on the «basic level» of the incentives and earned privileges scheme. (politics.co.uk)
The AAdvantage cards will not earn you this privilege. (upgradedpoints.com)