You already know that $\displaystyle \begin{align*} A = \frac{1}{2}b\,h \end{align*}$.

But if you look at the left-hand right-angle triangle, we can see that from the point of view of angle C, the opposite side is h and the hypotenuse is a.

So $\displaystyle \begin{align*} \sin{(C)} = \frac{h}{a} \implies h = a\sin{(C)} \end{align*}$.

Thus $\displaystyle \begin{align*} A = \frac{1}{2}b\,h = \frac{1}{2}a\,b\sin{(C)} \end{align*}$.