Let's come up with some more explicit descriptions of these sets:
$A=\{1\}$ $B=\{-9,-8,-7,-6,-5,-4,-3,-2,-1,0\}$ $C=\{\ldots,-3,-1,1,3,5\ldots\}=\{\text{odd numbers}\}$
First, do you understand why the above statements are true? If not I can explain in more detail.
Can you describe $A\cap B$? Then $C\setminus (A\cap B)$ consists of every element of $C$ that isn't in $A\cap B$ (hint: there is a very easy description of this set using $C$).
The second part is a bit trickier. $B\cup C$ consists of every element of either $B$ or $C$, so it is the odd numbers, together with $-8,-6,-4,-2,0$ (the odd numbers $-9,-7,-5,-3,-1$ are in both $B$ and $C$, but they don't get counted twice or anything). Then, $(B\cup C)\setminus A$ consists of every element of $B\cup C$, just throwing away anything in $A$.
EDIT: We can describe the set explicitly as $(B\cup C)\setminus A=\{n\in\mathbb{Z}: n=2k+1,k\in(\mathbb{Z}\setminus\{0\})\}\cup\{-8,-6,-4,-2,0\}$ The first set is just the odd numbers, except for 1 (which we threw out because it was in $A$), and then the second set is the even numbers we have to add in from $B$.