How does Telemachus feel after Athena leaves in the Odyssey?

When Athena first appears to Telemachus, the young man is not in a happy place. He's lost without his father, loudly bemoaning his fate to anyone who'll listen. But Telemachus needs to be roused from his self-pitying apathy. The onerous responsibility falls to Athena, who comes to visit Telemachus in the guise of Mentes, a Taphian lord. It's a common trope in Greek mythology for a god to take on the appearance of a harmless stranger as Athena does here. The disguised goddess patiently listens to the young man as he pours out his heart to her. Athena is the goddess of, among other things, wisdom and war, and Telemachus is going to need to draw on her skills in both those areas if he's going to drive out the suitors from his father's palace, as Athena encourages him to do.

Athena's trying to get Telemachus to put away childish things and make the step up to manhood. She knows full well that Odysseus is still alive and well, but she plays upon his son's uncertainty concerning his fate to get him to man up and participate in the elaborate plan she has devised. Telemachus is now full of purposeful courage, ready to take his first steps into the wider world as a man, the proud son of a great king and warrior. Athena has set the young man on his path, and so she leaves him for now. Taking to the skies, Athena can be satisfied that the first stage of her plan is now complete.

"Off and away Athena the bright-eyed goddess flew

like a bird in soaring flight

but left his spirit filled with nerve and courage,

charged with his father’s memory more than ever now,

He felt his sense quicken, overwhelmed with wonder—

This was a god, he knew it well and made at once

for the suitors, a man like a god himself."

Athena will continue to appear to Telemachus in the guise of Mentor—encouraging him, recruiting sailors for his forthcoming voyage, and dazing the suitors. But Athena's most important work is done. When Athena flies off, Telemachus has turned from a boy into a man. He feels confident in himself and in his own abilities, and his new-found self-belief gives him the courage to confront the suitors and address the assembly without fear before embarking upon his epic voyage.