Consider the log hazard, $\eta(t|X)=\eta_0(t) + X\beta$
and coding different groups using indicator variables with reference-cell coding.
Note then that $\eta_0(t)$ is the log-hazard function for the reference group and that the other groups have a log-hazard that is shifted up or down by some $\beta$ (or combination of $\beta$s).
A shift in log-hazard is the same as a scaling of the hazard function by a multiplicative constant, which is exactly what the name "proportional hazards" implies (that the groups' hazards are simply scaled versions of each other).
So this formulation using the $\exp$ is the natural way to get the log-hazards to be shifted (i.e. proportional hazard).
Now it is not necessarily the case that the model should be linear in $X$ on the log-hazard scale, that's an assumption, just as it would be in say a GLM (where it's not necessarily the case that the link-transformed mean $\eta=g(\mu)$ is linear in $X$, but it's an assumption that is often made).