"I guess the hardest part of moving on
Are these memories............."
"Moving on" is an expression that means to go ahead, to leave the past behind. We see it used when a couple breaks up or when a person dies, or when something happens. People will say, "I have to forget the past and move on." or "I got fired from that job, but I have to move on."
This person is saying that the most difficult thing to do (when you decide to "move on") is to forget all the memories of the past or pack them away. But the memories are strong and full of emotion, and they keep coming back for us to remember. This sometimes prevent people from "moving on".
It takes courage to "move on."