What is a “database-centric web-based” application mean for APEX?

It’s an application that runs in a browser and presents and updates data that resides in a database. In the case of Oracle APEX the data resides in an Oracle Database – specifically, the database in which APEX is installed.

APEX applications can access data in remote Oracle databases and non-Oracle databases, but the main, typical and normal data store is within the Oracle database in which APEX is installed.