Blowing your cover generally means revealing an identity (For instance, if a cop who is under cover reveals that they are a cop, they have blown their cover). It would make sense in that the person in the song feels like breaking out and showing the world who he actually is, though he may never do it.
The song is about someone getting up to do a job that he doesn't enjoy and watching his true desires get further and further out of reach; "But it's too late for me now" "But it's too late for me.... now". But that also ties in to a deeper major theme on Awake, regarding the notion of taking control of where you want to go and who you want to be rather than just riding the current, or getting caught in the cycle of trying to please everyone else. That can apply to creating music, finding companionship, or anything else. The "sound" in 6:00 seems to be a calling he has in his head to pursue a dream.