From elementary probability course, the probability distributions such as Gaussian, Poisson or exponential all have a good motivation. After staring at the formula of the exponential family distributions for a long time, I still do not get any intuition.
$$f_{X}(x\mid {\boldsymbol {\theta }})=h(x)\exp {\Big (}{\boldsymbol {\eta }}({\boldsymbol {\theta }})\cdot \mathbf {T} (x)-A({\boldsymbol {\theta }}){\Big )}$$
Can anyone help me understand Why we need it in the first place? What are some advantages of modeling a response variable to be exponential family vs normal?
EDIT: By the exponential family, I meant the general class of distributions described here.