This stage of the hero's journey is really all about fear, internal and external.
You've been asked to take on a big adventure, to do something you've never done before - a trip up Mt. Kilimanjaro or a weekend sailing trip or even something as simple as going to yoga or the gym for the very first time. You've said yes but now it's time to pack up and actually leave home. And you're scared to death. You think of all the ways you can get out of it. I have to work this weekend; it's my mother's birthday and I forgot; I've got a touch of the flu. You may even make the phone call to pull out. This is what the refusal of the call is all about.
Because this stage is one of the places where we learn that the hero is human, this is one of the places where we begin to identify with the hero. Because - as I've shown above - we've all been in this situation. We've all been invited to try something new, to take a risk, and we've all contemplated backing out of it. Because all of us are scared of the unknown.
It doesn't matter whether the unknown is a big thing (Frodo and Samwise on their journey across Middle Earth to throw the ring into the fire) or a small thing (your very first day at a new job or your first writing class or conference), we all feel that fear of the unknown. And sometimes, even the bravest of us, want to back out.
This fear can be external - the hike up Mt. Kilimanjaro or the sailing trip - or internal - a new job or a blind date. It doesn't matter. Because we've been there, we identify with this fear of the unknown.
For me, whether I'm writing, watching a movie, or reading a book - this stage is absolutely crucial. I want to know that the hero is as scared as I am about new things. I want to see the hero overcome that fear because then I believe that I can overcome it as well. I want to know that the hero can make mistakes, can know that the task must be undertaken (like Frodo and the ring, me and a phone call to someone I don't know), and still be scared to death to do it.
Story for me never works unless I can identify with the hero - so if they're too perfect, I'm not interested. This is where I get to know the hero, where I can (no matter how big or small the task) put myself in their shoes.
This is where the story takes off for me. I may have been interested before this point, but when this happens, I'm there. So make sure that the quest and the hero's refusal of it are clear. Don't be wishy-washy at this stage - go for it. This is where your reader is going to buy into the hero, this is where they're going to say, oh, I definitely have to see what happens here.