Eigen Function of Linear Time Invariant (LTI) System
You can use the Convolution Theorem to show that $ \int_{-\infty}^{\infty} h \left( \tau \right) x \left( t - \tau \right) d \tau $ is equivalent to Element Wise multiplication in the Frequency Domain.  

Hence for the case $ x \left( t \right) = A {e}^{s t} $ where $ s \in \mathbb{C} $ the output is just a multiplication by scalar of the input (The value of the System Transfer Function at the Delta point of the input).

See Eigen Functions of LTI Systems.