Example 4 As shown in Figure 6, in $\triangle A B C$, it is known that $\angle B A C$ $=\angle B C A=44^{\circ}, M$ is a point inside $\triangle A B C$, such that $\angle M C A=30^{\circ}$, $\angle M A C=16^{\circ}$. Find the degree measure of $\angle B M C$.