In Book 4 of the Odyssey, how does the appearance of the the phantom support the idea that this story is an epic?
I assume that you are talking about the episode at the very end of Book IV where Athena creates a vision in the form of Penelope's sister and makes that vision appear to Penelope. The vision tells Penelope to stop worrying because Athena is with Odysseus and is protecting and helping him.
Acccording to the literary-terms/epic link below, one of the characteristics of an epic is that it involves "superhuman forces entering the action." This is the part of the definition of an epic that is relevant to this question.
In this passage, a superhuman force is intervening in the action of the story. Athena, as a goddess, is clearly superhuman. She is participating in the action by helping Odysseus and by, in this case, helping Penelope.
The phantom, as you call it, is an example of superhuman forces participating in the action and that is one of the distinctive characteristics of an epic.