Let $Z\sim N(0,1)$ and its mgf $m(t)=e^{{t^2}/2}$. The third moment of Z is the third derivative of the mgf: $m^3(t)=t^3 e^{{t^2}/2}+2te^{{t^2}/2}...$ at $t=0$.
Why is $t$ set to zero? Is it question specific? I never really understood what the $t$ was, I just thought each distribution came with a specific value of $t$ that would make the mgf "work" for it...
There are really two questions here, each warranting a paragraph.
The mgf $m(t)$ is a function of a dummy parameter $t$, just as the CDF $F(x)$ is a function of a dummy parameter $x$. They're two different ways to describe the distribution's degrees of freedom with a function.
Since $m(t)=\Bbb E\exp tX$, the $n$th derivative $m^{(n)}(t)=\Bbb EX^n\exp tX$, so $m^{(n)}(0)=\Bbb EX^n$. This use of $t=0$ is applicable to any distribution for which the MGF is differentiable $n$ times at $0$.