The answer to this question is closely connected to the answer to the last question you asked. As I said before, Nestor has suggested that Telemachus should go to Sparta and ask Menelaus about Odysseus's whereabouts. Lacedaemon is another name for Sparta. So by telling Telemachus to do this, he is telling him to go to Sparta.
The reason Nestor wants his sons to go with Telemachus is because it will be safer for Telemachus. Nestor says it would be okay for Odysseus to go by sea, but it seems clear that he thinks it would be safer to go by chariot with Nestor's sons to guide him.
Nestor will not allow Telemachus and Athena (though he doesn't know it's Athena yet) return to their ship without eating and resting in comfort in his home. The ancient Greek concept of xenia translates to hospitality, and since Zeus was the protector of travelers, it was considered a religious imperative to offer great hospitality when travelers came, in need, to one's door. Nestor offers such hospitality to Telemachus, and Athena thanks him heartily and says, "'As for Telemachus who stays with you, send him upon his way by chariot with your son, and give him horses that have swiftest speed and best endurance.'" Immediately after this, she changes into an osprey and flies away; in so doing, she has shown that she is an immortal, and Nestor isn't about to disregard something a goddess has told him to do. He'd already offered his sons to be Telemachus's guide to Lacedaemon, but this cements his resolve.