Comprehend it this way: You create a blank (A)GPL project and incorporate the upstream BSD codebase into it. While those original upstream files remain under their original permissive license, the project as a whole is governed by the (A)GPL (plus the attribution requirements of the upstream license, which the GPL permits). From there, you can add your own code under the AGPL and distribute the combined work under the AGPL.
If someone takes your code and uses only your portion, they can use it under the AGPL alone. However, if they also include the upstream source code, then the attribution requirements of the upstream license must still be met.