Step 1: Test Assertion (I) using the standard variance decomposition. For any population, the total variance can be split as $\sigma^2 = \sigma_w^2 + \sigma_b^2$, where $\sigma_w^2$ is the (weighted) within-stratum variance and $\sigma_b^2$ is the between-stratum variance. The variance of the stratified mean (proportional allocation) depends only on $\sigma_w^2/n$, which is at most $\sigma^2/n$, the SRS variance. This proves stratification (properly done) never increases sampling error, confirming (I) is true.
Step 2: Test Reason (II) independently. Given a fixed $n = \sum n_h$, minimizing $Var(\bar y_{st}) = \sum \dfrac{N_h^2 \sigma_h^2}{n_h}$ subject to that constraint by Lagrange/Cauchy-Schwarz gives the Neyman solution $n_h \propto N_h \sigma_h$, and this minimized variance is mathematically less than or equal to the variance under proportional allocation. So (II) is also independently true.
Step 3: Judge the explanatory link. (I) speaks generally about "following the principle of stratification" (which includes even equal or proportional allocation, not necessarily optimal), while (II) is a special, stronger claim restricted to optimal allocation. Since (I) would remain true even without invoking optimal allocation, (II) cannot be regarded as the reason behind (I); the two are true but logically independent statements.
So both statements hold, yet the reason given is not what explains the assertion.
\[\boxed{\text{Both (I) and (II) true, (II) is not the correct explanation of (I)}}\]