Let's take the spin operator along a unitary vector  $\hat{\vec \sigma} \cdot \vec n $. We can find the spin eigenstate with eigenvalue $\hbar/2$. If we rotate the system, $\vec n$ is transformed by a $SO(3)$ matrix and (since the spin eigenstate depends on $\vec n$) the spin eigenstate transforms consequently. If we search the matrices that transforms the spin eigenstate as described above, will we find the matrices of $SU(2)$?