Discuss ways in which Homer uses everyday details of Greek life to enhance his tale and give it a familiar feel for his audience.

One key way that Homer employs everyday details in telling his stories is through comparing great actions to day-to-day ones through extended metaphors. (Important terms to know about metaphor include tenor, that is, the literal situation that the writer or speaker is describing, and vehicle, that is, the situation to which it is being compared.) A normal metaphor would be something like "her boyfriend is a clown." The boyfriend (the tenor) is not literally making a living as a clown (the vehicle.) However, he is like a clown because he is a fool or is very funny, depending on the situation. If I were to say, "Her boyfriend is like a clown who trips over his own feet, knocks someone else over, and runs away laughing," you would know that I meant he was a fool. This is an example of an extended metaphor; it is when the writer or speaker expands on the situation described in the basic metaphor to tell you more about the item or situation that is the tenor.
To return to Homer: Homer uses extended metaphors in which the great happenings of the Trojan War or of Odysseus's heroic journey are the tenor, and everyday details are the vehicle. For example:

As ravenous wolves come swooping down on lambs or kids
to snatch them away from right amidst their flock, all lost,
when a careless shepherd leaves them straggling down the hills
and quickly spotting a chance the wolf pack picks them off,
no heart for the fight, so the Achaeans mauled the Trojans.

Homer's hearers might not ever have seen an army destroying another army, but they would be very familiar with the situation Homer describes involving the wolves, the lambs, and the shepherd. This detailed description of a familiar event would allow them to envision something they had never seen before. It is not that the extended metaphors and the daily details they involve make the story itself feel familiar to the listeners; rather, Homer makes use of the familiar to engage his audience's imagination and allow them to see the grand events he is describing—to "enhance his tale," as you say.