In The Odyssey, why does the arrival of Philoetius hearten Odysseus in Book 20?
Let us just remind ourselves of the situation that Odysseus faces. He has returned to his kingdom of Ithaca to find his house full of suitors who are eating away his wealth and property and trying to convince his wife to marry one of them. In his disguised state, Odysseus hopes to find help and any sign of the loyalty that he is rightfully owed as ruler of Ithaca. However, up until this stage in the narrative, he has found little evidence of this. The loyalty of Philoetius to his master is something that gives him great encouragement and joy especially when he is compared to the character of Melanthius, who treats Odysseus in a terrible, disrespectful way. Note how Philoetius talks to Odysseus (not realising his true identity):
A sweat came over me when I saw this man, and my eyes filled with tears, for he reminds me of Ulysses, who I fear is going about in just such rags as this man's are, if indeed he is still among the living.
The loyalty and love of Philoetius towards his master gives Odysseus great hope and courage at this point in the narrative, when so many things seem to be against him. To have one ally is better than having none at all.