Even with the "intelligent but unfamiliar" approach, it can be difficult to figure out how much explaining you need to do in a paper. As long as your professor doesn't have any rules against it, trying out a draft on a friend or even a tutor in the writing center can be a good way to get an idea of how clear you've made things. If your reader has questions or is confused by the paper, you know you probably need to include more explanation. But if you've provided your reader with too much irrelevant detail, they may get bored. Ultimately you want a balance between relevancy and coverage. Try to aim for enough detail and explanation to support the points you make throughout your paper and give your reader a basic understanding of the topic.
Writing can be a little bit like performing, in that you do it for a specific audience and you need to keep that audience in mind as you work, otherwise you might end up with a final product that is completely inappropriate for your audience. Think about the first party you went to in college. How would you describe that party to your best friend? How would you describe it to your mother? What about your grandmother?
It's possible that you're totally innocent and angelic and have nothing to say that would shock granny, in which case, congratulations and keep up the good work. Chances are though, your story is going to change a little depending on which person you're telling it to. Maybe some of the details will be a little different, or maybe you'll stress some things and downplay others, or even leave some out all together. Believe it or not, you should approach writing an essay in the same way.
Technically speaking, your professor is your audience. She or he is the one who will be reading and grading your assignment, so you definitely want to take that into account while writing. But if you focus on your professor too much, you might leave out some valuable information. After all, your professor probably knows more than you do about whatever topic you're working on right?
Wrong! Ok, sure, your professor probably does know more than you do, but if you base your writing on that assumption, if you don't fully and clearly explain your points and all related information, it's going to look like you don't completely understand the topic. Instead of thinking of your professor as your primary audience, imagine that you're writing to someone intelligent, but who is uneducated in your topic area. Include enough basic explanation for someone who isn't familiar with the topic to understand.