Let's say I have an FIR filter with the equation:
$$ y[n] = \sum_{i=0}^{N-1} h[i] x[n-i] $$
I know that to find the frequency response of this filter, I need to input a complex exponential in place of $x[n-1]$
$$ x[n] = A e^{j\phi} e^{j \hat{\omega} n} $$
and the resulting frequency response will be
$$ H(\hat{\omega}) = \sum_{i=0}^{N-1} h[i] e^{-j \hat{\omega} i}. $$
I understand $e^j$ to mean rotating around in a circular motion, which is a convenient way to represent complex sinusoids.
However, I don't understand how inputting this general form of a complex exponential into this system works to give the frequency response of the system. How can I understand this intuitively?