Forging raises a relic's base numbers by one rank. Think of putting it back on the anvil. The option stays as it is and attack, defense and health go up.
You do it on the Gear tab of the mercenary sheet. Both the worn relic card and the warehouse relic card carry a "Forge ★1 · 60 shards" button.
Each rank costs another 60. You can forge up to one rank below your best generation, and higher ranks wait for the next generation.