In our civilized, modern life, we use our teeth for eating only, and our fingernails and toenails are used hardly at all, except for decoration. Among animals, however, teeth and nails are often used to collect food or catch prey. More aggressive animals, such as tigers and bears, use teeth and nails to attack other creatures or defend their territory. If attacked themselves, they will fiercely "fight tooth and nail" to avoid being wounded or captured.

From that animal origin, we have imported this idiom into our civilized, modern life to represent any situation where we humans feel we must fight aggressively for some cause or purpose. It does not mean to fight physically, as animals do, but to fight verbally using ideas and arguments.