There are three reasons why the meeting between Telemachus and Nestor is important to the development of the work (and Telemachus). First, Telemachus gets a better understanding of what happened at Troy and more importantly the aftermath of the war. Second, he gets encouragement to seek out his father. This is an important point, because he is still relatively young and he needs encouragement. With the encouragement of Nestor, he sets out to Sparta.
At this point, we need to remember what Nestor is known for. He is a wise and persuasive speaker. In light of this, Nestor gives the story of Orestes, who was a faithful son to Agamemnon. By this example, he encourages Telemachus to be the dutiful son that he should be.
Finally, Nestor sends his own son along with Telemachus on a voyage to Sparta. This companionship between two sons is what Telemachus needs at this juncture of his character development.
The Odyssey is one of the most famous epic poems ever created. While attributed to the Greek poet Homer, we do not truly know who first imagined what within the story--because at the end of the 8th century BC, poems were spoken aloud rather than written down. Homer could have been one person, a group of people, or even just an imaginary figure given credit for orally passed down poems.
The epic poem details Greek hero Odysseus' ten year journey home to Ithaca, where he is King. While he encounters various hurdles such as monsters and nymphs, his wife (Penelope) and son (Telemachus) are waiting at home and dealing with suitors who want to marry Penelope and take Odysseus' place.
To put the time into perspective: Odysseus was at war for ten years. He then took ten more years to get home. That is twenty years that Ithaca lives without their king. The people of Ithaca grew restless and wanted Penelope to remarry, because they assumed that Odysseus had died on the journey home. Nestor, who fought alongside Odysseus in the battle of Troy, made it home safely, so Telemachus seeks him to find out any information he can about his father.
Telemachus goes to Pylos with Athena, who encourages him to speak to Nestor. He asks about his father, whose whereabouts are unknown, and who might possibly be dead. Telemachus is initially positive that his father must be dead, and he complains of the suitors who wreak havoc upon his house in their quest to marry his mother. Nestor dismisses this and encourages Telemachus to go to see Menelaus and ask him what he knows about Odysseus. Nestor also offers to have his sons lead Telemachus to Menelaus.
This encounter is significant for two reasons: Telemachus is given hope, and Telemachus is given encouragement. Without the assurance of Nestor that hope should not be lost, Telemachus might have buckled under the pressure of the suitors. However, Nestor suggests that Odysseus might still make it home. Nestor also provides a momentary father figure to Telemachus, who has lived without one since his father has been at war for many years. Nestor not only tells Telemachus to keep his hope, but also gives him the aid of his sons so that Telemachus can successfully reach Menelaus.
This is a moment of growth for Telemachus, who has only lived with his mother up to this point. Nestor is able to give him inspiration to search for his father and not accept that he is dead. Nestor also offers the advice that Telemachus should not stray far from Ithaca, because if he does, the suitors might be able to advance on his mother. Without Nestor, Telemachus might have lost hope in his father's life, and might have remained a fearful man instead of the motivated man he becomes with inspiration.