Arguably, Bilbo shows courage just by deciding to go on the quest in the first place. In Tolkien's mythos, hobbits are the everyman figures; they're homebodies who appreciate the good things in life (food, parties, etc.), and they don't really welcome change or uncertainty. For Bilbo, then, courage tends to be about pushing past his instinctive fears and reservations, and he does this throughout the novel—for instance, when he confronts the trolls in chapter 2, or (even more impressively) when he enters Smaug's lair in chapter 12. Bilbo is frightened enough in both instances to berate himself for ever agreeing to go on the quest, but his sense of duty and loyalty ultimately proves stronger than his fear.
With that said, the kind of courage and leadership Bilbo displays never quite map onto our conventional definitions of either. In literature—and especially in genres like fantasy or medieval romance—being a hero tends to mean being exceptional in some way (strength, power, wisdom, etc.). This, of course, is a better description of a character like Thorin than it is of Bilbo, and yet Tolkien portrays Thorin as deeply flawed (if still sympathetic). Thorin is brave and noble, and he generally observes a very strict behavioral code (think, for instance, of the formality of the contract he asks Bilbo to sign). Ultimately, however, he is much more susceptible to the corrupting influence of wealth than Bilbo. By contrast, Bilbo plays somewhat fast and loose with the rules; he's very good, for instance, at talking himself out of situations where a more traditional hero would fight (e.g., his interaction with the trolls, or the riddle game he plays with Gollum). This ability to think on his feet and bend the truth when necessary ultimately wins him the respect of the dwarves; by the time the group reaches the Lonely Mountain, Bilbo "has become the real leader in their adventure," with the dwarves looking to him for plans (chapter 12).
More even then cleverness, though, Tolkien suggests that it is Bilbo's humble and grounded sense of morality that makes him brave and a good leader. He steals the Arkenstone, for instance, not to keep it for himself but rather in an effort to prevent war (and save his newfound friends in the process). Thorin casts him out of the Lonely Mountain in response but ultimately acknowledges that Bilbo was in the right, suggesting that Bilbo's "courage" and "wisdom" spring in part from his valuing "food and cheer and song above hoarded gold" (chapter 18). Perhaps most significant, however, is Bilbo's decision to spare Gollum's life at a moment when it might have been safer or more prudent to kill him; although Gollum was blocking Bilbo's path out of the goblin tunnels, Bilbo's "pity" for Gollum prevented him from harming him (chapter 5). Tolkien's work was deeply influenced by his Christian worldview, so it is Bilbo's ability to show compassion, even at great risk or sacrifice to himself, that ultimately makes Bilbo a true hero.