Why training error is larger than validation error after PCA?
The basic principle that helps us implement PCA is it's ability to explain the variation, so it depends how much PC you are taking and how much variation they are explaining. They are definitely not explaining 100% of variation so there are good chance that Validation Error is higher.

Before applying PCA you need to understand why we apply it? We apply it to reduce computational effort and practical implementation of problem. The idea of PCA is not to give better result than actual variables in problem.

Better results are generally available with complete list of variables.