One driving motivating factor is that Aeneas was the son of the goddess Aphrodite by the Trojan royal, Anchises. Demi-gods and those of royal blood are traditionally thought to be of superior courage, foresight, power, and drive. So with both strains of blood flowing through his veins, Aeneas is doubly superior in these qualities and thus doubly motivated. In addition, he would doubly understand and believe and attend to the words of other gods, such as Jupiter's prophesy that Aeneas would found a great empire.
Aeneas is a character who above all believes in the gods and their call on his life in order to fulfill a specific purpose. It is this divine awareness that gives him the strength to carry on in his task no matter how hard it gets. Aeneas is therefore a character who is defined by his piety, or belief in the gods, as the way that he lives his life shows.
Aeneas himself, in the by A. S. Kline, explains his departure as follows:
it is Italy that Apollo of Grynium,
Italy, that the Lycian oracles, order me to take:
that is my desire, that is my country.
Later Virgil adds that
dutiful Aeneas, though he desired to ease her [Dido's] sadness
by comforting her and to turn aside pain with words, still,
with much sighing, and a heart shaken by the strength of her love,
followed the divine command, and returned to the fleet.
Thus, "divine command" -- obedience to the gods -- is the reason that Aeneas himself offers.
There are a number of things that we can say. Let me mention two of them. First, Aeneas believes in the words of Jupiter. There is a prophecy that the gods would enable him to find a land and give his people rule without end (imperium sine fine). So, no matter how hard things are, he persists. Second, Aeneas is characterized by piety (pietas). In the Roman world this is best defined as being faithful to a person's obligations or duty. Aeneas, therefore, is duty bound to establish a new city. For this reason, he persists no matter what the cost. Finally, if we were to give another reason, we can say that he loves his people and wants them to have a new land.