It's an unfortunate occurrence for all anglers - sometimes fish swallow the bait resulting in your hook point being buried deep in the gullet with little chance of safe removal through conventional means. We owe the fish and resource our best effort at clean extraction.
The safest and most expedient method for a clean removal is rolling the hook backward via needle nose entry through the gills. Wired2Fish's Mitch Anderson demonstrates the procedure that easily removes the deepest hooks with little tear or damage to the fish beyond the initial hookset and fight. Let 'em go to grow and fight another day with a high probability of survival!