The house of Odysseus has been infiltrated by the suitors. They are steps away from staging an all-out revolt to force Penelope to marry, thus gaining the riches of the household. If Telemachus were to leave during the day, everyone would know he was gone. This would weaken the small hold Penelope has in keeping the suitors at bay. To leave at night keeps his secret. Also, Penelope would disagree with Telemachus' decision, and he does not want to argue with or disobey her.
Telemachus sails away in the night to find more information about his father Odysseus because Athena advises him to. Athena, the goddess of war and wisdom, is also the special guardian of Odysseus and his family. She wants to help him get home, and she hopes that his son Telemachus will aid in that.
They have to leave in the night because of the suitors and because of his mother Penelope. If the suitors knew he was trying to leave, they would kill him. They are already frustrated because of Penelope's tricks to hold them back, and they are afraid of anything that might delay a possible marriage. Penelope, on the other hand, is terrified that Telemachus will never return. She has already suffered through one loved one going out to sea and not returning, and she cannot bear the thought of the same thing happening to her precious son. She definitely does not want him sailing off anywhere. They have been arguing a lot lately, over the suitors and over her steadfast hope that her husband will return. She also does not want to left alone to deal with the suitors.
Thanks for your help :)