From the Catena Aurea:
Augustine, de Trin., 4, 13: Why did He offer Himself to temptation? That He might be our mediator in vanquishing temptation not by aid only, but by example.
Chrysostom: But that you may learn how great a good is fasting, and what a mighty shield against the Devil, and that after baptism you ought to give attention to fasting and not to lusts, therefore Christ fasted, not Himself needing it, but teaching us by His example.
Augustine, Serm. 210, 2: Not however because Christ fasted immediately after having received baptism, are we to suppose that He established a rule to be observed, that we should fast immediately after His baptism. But when the conflict with the tempter is sore, then we ought to fast, that the body may fulfil its warfare by chastisement, and the soul obtain victory by humiliation.
Hilary: And therefore in the temptation [the devil] makes a proposal of such a double kind by which His divinity would be made known by the miracle of the transformation, the weakness of the man deceived by the delight of food.
Jerome: Christ’s purpose was to vanquish by humility;
Leo, Serm. 39, 3: hence he opposed the adversary rather by testimonies out of the Law, than by miraculous powers; thus at the same time giving more honour to man, and more disgrace to the adversary, when the enemy of the human race thus seemed to be overcome by man rather than by God.