Odysseus is brave because he repeatedly faces mythological creatures with resolve. He never backs down from a challenge, and he always keeps a level head in dangerous situations.
Odysseus faces the Cyclops Polyphemus, travels to the Underworld, overcomes the temptation of the Sirens, outmaneuvers the monsters Scylla and Charybdis, and leaves behind the beautiful Calypso. All these feats required a great amount of bravery and skill, two traits Odysseus was honored for in antiquity.
His optimism that he will return home to his wife safely after 10 years of war (The Iliad) and 10 years of travel (The Odyssey) are further signs of his innate bravery and confidence.