tl;dr

Actually, it is pretty easy.

db.books.aggregate([
    { $match:{ Category:"Children" }},
    { $project:{ _id:0, author:1 }},
    { $unwind:"$author" },
    { $group:{ _id:"$author", count:{ $sum: 1} }},
    { $sort:{ count:-1 }},
    { $limit: 10 } 
])

In detail

Sample data

{ "_id" : ObjectId("570ec0a29307105e18d42bae"), "author" : [ "Mary", "Tony" ], "booktitle" : "Book1", "Category" : "Children" }
{ "_id" : ObjectId("570ec0cf9307105e18d42baf"), "author" : [ "Joe", "Tony" ], "booktitle" : "Book2", "Category" : "Children" }
{ "_id" : ObjectId("570ec0f59307105e18d42bb0"), "author" : [ "Joe" ], "booktitle" : "Book3", "Category" : "comedy" }

The aggregation explained

1. { $match:{ Category:"Children" }} First we find all books in the category in question
2. { $project:{ _id:0, author:1 }} We only want the author array, which we

3. { $unwind:"$author" }. That means that for each array element we create a new document, which retains the other fields of the original document. In our case, this is none, and hence, at this stage the result of our aggregation looks like this:

   { "author" : "Mary" }
   { "author" : "Tony" }
   { "author" : "Joe" }
   { "author" : "Tony" }
   

4. { $group:{ _id:"$author", count:{ $sum: 1} }} Next, we simply group by author and increment a counter for each occurrence of the same distinct author name.
5. { $sort:{ count:-1 }} We sort by the field we just created, count, in descending order and
6. { $limit: 10 } limit to the ten authors which have the highest count

The aggregation gives the following

Result

{ "_id" : "Tony", "count" : 2 }
{ "_id" : "Joe", "count" : 1 }
{ "_id" : "Mary", "count" : 1 }