He is the Bread of Life, through whom we are promised to never hunger or thirst. Clearly it would be overly simplistic to assume he means here physical hunger and thirst, particularly in light of how he opens the discourse (and the simple fact that his closest followers hungered and thirsted on a regular basis!). Rather he is pointing to a deeper hunger, a deeper thirst here; that hunger and thirst in our hearts that cannot be sated by all of the pleasures in the world or even the world itself, but that He promises to satisfy entirely.
How does he do that? He gives us Himself, the Life and the Love that He Is. He who is the Bread of Life emptied himself on the Cross (Phil 2:5-8) so that we might be filled with him—body, blood, soul and divinity—in the Eucharist.