What you are doing, in terms of the license, is:
- You are modifying the software. Then you must comply with section 13 of the AGPL.
- You are "conveying" the modified version. Then you must comply with section 5 and/or the following section (6) depending on whether you're "conveying" source or binaries.
Let's start with sections 5 and 6. Basically, these sections say that, for each person or organization who receives the modified software from you, you have to do all of the following:
- License the software to them under the terms of the AGPL, and tell the recipient that you have done so.
- Provide source code, either because you're only conveying source code in the first place, or in addition to binaries/object code.
- Tell the recipient that you modified it, and provide a copy of the original copyright notice as well as any similar legal notices that were present in the original.
- If the original has a UI, and the UI can display a notice telling the user about the copyright status of the software, you must not remove or disable access to that UI element. You may modify or rearrange the UI, but the element must still be accessible somewhere, unless you have completely removed the UI altogether (e.g. because you turned a complete program into a library).
- If the software is intended for use in a particular piece of consumer hardware, and sold together with that hardware as a single transaction, then you must provide the recipient with the practical ability to run modified versions of the software on the hardware, unless the hardware is designed in such a way that nobody could possibly run modified software on that hardware (e.g. because it's physically burned into ROM).
- You can't charge an additional fee for any of the above as separate add-ons, but you may charge a base fee for the entire transaction as a whole. In practice, this is rarely commercially feasible because your clients can immediately turn around and give your software away for free, but if you can find a way to make it work, the license allows you to charge such a fee, as long as you don't charge extra for exercising the rights listed above.
These conditions are more or less the same as those of the regular GPL. Note that the above is a simplification of the actual legal requirements. You should read the license for a full understanding of exactly what these terms mean, or consult a copyright attorney.
Importantly, none of these requirements apply to anyone other than you and the recipient. If some third party wants access to the source code, for example, that's their problem (but see below!).
Now, let's move on to section 13, which has no GPL equivalent. Section 13 is short enough that we can quote the relevant paragraph in its entirety:
Notwithstanding any other provision of this License, if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software. This Corresponding Source shall include the Corresponding Source for any work covered by version 3 of the GNU General Public License that is incorporated pursuant to the following paragraph.
In English, this is saying that, if your software has the ability to be remotely used over a network (e.g. a web application, a server, etc.), then your modified version of the software must make it practically possible for the client (the person at the other end of the network) to receive a copy of the modified source code, for free.