Why are we using exp(.) as the functional form in the household maximization problem inside the RCK model?
I guess you already know that $\exp(-\rho t)$ is the the discount factor. In particular, it is obtained by continuously compounding the discount factor

$$\exp(-\rho t)=\lim_{m \to \infty} \left(1+ \frac{\rho}{m}\right)^{mt}$$