If I have a function like $F(x,y,z)=xyz-\cos(x+y+z)$, I am asked to use implicit differentiation to solve for $Dz/Dx$, and $Dz/Dy$.
I want to know why
$$\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$$ $$\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$$
instead of;
$yz+\sin(x+y+z)\cdot(1)$
for example.
When do I use this theorem?
Suppose we are looking at the solution of $F(x,y,z)=c\in \mathbb{R}$ for some fixed $c$.
Under reasonable conditions on $F$, this locally defines a smooth surface in $R^3$ (the implicit function theorem gives this), and so at most points on this surface, we can write any coordinate as a function of the other two, eg $z=z(x,y)$.
Now since $F$ is constant on our surface, the chain rule tells us
$$0=\partial_x(F(x,y,z(x,y)))=F_x(x,y,z(x,y))+F_z(x,y,z(x,y))\cdot \frac{\partial z}{\partial x}$$ and $$0=\partial_y(F(x,y,z(x,y)))=F_y(x,y,z(x,y))+F_z(x,y,z(x,y))\cdot \frac{\partial z}{\partial y}.$$
Solve these two equations for the partial derivatives of $z(x,y)$ to get exactly what you want.
Note that you will not necessarily be able to write $z$ as a function of $x$ and $y$ at points where $\partial_zF=0$, and also that there is nothing special about $n=3$ here, this kind of calculation is valid in any number of dimensions.