When a project on GitHub uses two OS licenses, Apache/EPL or Apache/AGPL, does one license govern over the other or do both govern equally and together? My main confusion is how the two licenses interact.
Generally, this is called
Multi-licensing when you want a particular product to have more than one condition. Any product can have more than a single license.
In Open Source it has its own term which is called
dual-licensing, and its use-case is similar to that of the general team
Multi-licensing, we do this when we want our open source project to have two separate conditions for our open source project to be used, if the Apache license is blocking some certain use-case that you would like to use the OS project for, you check for the second license if it supports it before using it the way you would like.
You can read this to understand better how
dual-licensing works in OSS.
As the owner of the copyright, it is your prerogative to allow others to do the things covered by copyright law (notably copying and distributing further) under whatever conditions you'd like. You can offer the recipient several alternative sets of conditions (i.e., use it under GPLv2, or under Apache, or..., or you are allowed --presumably for a fee-- to use it in ways the preceding don't allow).