How does Telemachus mature throughout The Odyssey due to the absence of his father?
With his father gone, Telemachus must step up and become the man of the house. He must grow into his power as a young man and direct the servants and even his mother. We see him issue orders to her. For instance, he tells her,
Tend to your own tasks,
the distaff and the loom, and keep the women
working hard as well. As for giving orders,
men will see to that, but I most of all:
hold the reins of power in this house.
We can see that Telemachus no longer acts like a little boy. He issues directives like this to his mother—and she finds him to be speaking "clear good sense"—and he also tries to keep the suitors in order (though he succeeds in outsmarting them, with Athena's help, he cannot muscle them out).
Further, Telemachus actually leaves home on a ship in hopes of finding some information pertaining to his father's whereabouts. This takes a lot of bravery and perseverance. Then, when Odysseus does return, we see how much he trusts Telemachus and respects him, and the son is able to help the father reclaim his rightful place.