Expresses doing something with frantic, all-out effort, as if one's life depended on it. Often used with verbs of action, pleading, or escape.
He ran away for dear life.
I desperately called for help.
From 必死 (ひっし, 'inevitable death; desperation') + に + なって (te-form of なる, 'to become'), literally 'becoming desperate'. The phrase is a fixed adverbial expression.