'Your finest hour' refers to the moment in your life when you make your greatest acheivement or have your most success. In this sentence (taken from Radio Ga Ga if I am not mistaken), it means your moment of most success has not yet happened, and will happen in the future. Given the recent success of podcasts, maybe Queen were absolutely right!

"Fine" means "very good" so "your finest hour" means "the peak of happiness/success/popularity". It means that you haven't gotten happiness/success at peak yet so this success is yet to come to you and you are yet to experience it.

It means that whatever you are doing or trying to accomplish, you have not yet had your best ( whatever the task or goal is ) yet. The implication is that you will soon or eventually. So at your finest hour it means you are at the peak of your ability or performance of some sort of task or goal.