Why is this Conditional Density Function correct? This answered question shows how to solve the problem but I still do not understand how to get the conditional density function, i.e.
${"}$Let $Z=X+Y$, then the density $f_{X,Z}$ of $(X,Z)$ is defined by $f_{X,Z}(x,z)=f_X(x)f_Y(z-x)$ because $X$ and $Y$ are independent hence the conditional distribution of $X$ conditionally on $Z=z$ is proportional to $f_X(x)f_Y(z-x)$, that is, 
$$
f_{X\mid Z}(x\mid z)=\frac1{c(z)}f_X(x)f_Y(z-x),\qquad
c(z)=\displaystyle\int f_X(t)f_Y(z-t)\mathrm dt."
$$
I keep getting that 
$$
f_{X\mid Z}(x\mid z)=\frac1{c(z)}f_X(x)f_Z(z)=\frac1{c(z)}f_X(x)\int f_X(x)f_Y(z-x),\qquad
c(z)=\displaystyle\int f_X(x)f_Y(z-x)\mathrm dx.
$$
which just ends up being
$$
f_{X\mid Z}(x\mid z)=f_X(x).
$$
I know that I'm making a fundamental error, could someone please explain in more detail what it is? 

The general formula for conditional density is
$$
f_{X\mid Z}(x\mid z) = {f_{X,Z}(x,z)\over f_Z(x)}.\tag1
$$
Your error is in replacing the numerator in (1) with
$$f_{X,Z}(x,z)=f_X(x)f_Z(z),
$$
which is true only if $X$ and $Z$ are independent.
What you should do is replace the numerator in (1) with
$$
f_{X,Z}(x,z)=f_X(x)f_Y(z-x)
$$
and the denominator in (1) with
$$
f_Z(z)=\int f_{X,Z}(x,z)\,\mathrm dx = \int f_X(x)f_Y(z-x)\,\mathrm dx=:c(z),
$$
and you'll get the desired conditional density function.
