For the first result, notice
\begin{equation*}
M(0) = E[e^{0}] = E[1] = 1
\end{equation*}
is pretty trivial.
For the next results, take derivatives and use the linearity of the summation or integral. Here, let’s consider the case where \(X\) is a continuous variable and leave the discrete case for you.
\begin{equation*}
M'(t) = D_t \left [ \int_R e^{tx} f(x) dx \right ] = \int_R x e^{tx} f(x) dx
\end{equation*}
and then evaluating at t=0 gives
\begin{equation*}
M'(0) = \int_R x e^{0} f(x) dx = \mu .
\end{equation*}
Continuing,
\begin{equation*}
M''(t) = D_t \left [ \int_R x e^{tx} f(x) dx \right ] = \int_R x^2 e^{tx} f(x) dx
\end{equation*}
and evaluating at t=0 gives
\begin{equation*}
M''(0) = \int_R x^2 e^{0} f(x) dx = E[x^2] = \sigma^2 + \mu^2.
\end{equation*}
It should be noted that one may also determine the skewness and the kurtosis in a similar manner.