AGPL fixes a "bug" in GPL that many tech companies have exploited to not release modified source code as the GPL requires. Simply put, GPL says that you have to release the complete source code (including any changes you have made) of a GPL licensed code only if you distribute it to other users. Many tech companies thus avoided GPL code. But with the growth of Software-as-a-Service, where an application only runs on the server and is accessed through a browser or an app, many of these same companies created web applications with GPL code that they customised. However, if a user demanded the source code of the GPL code, along with the modifications they made, the tech companies refused to provide it claiming they weren't "distributing" the application (as in giving you the whole application to run on your computer). And since they weren't doing that they claimed they had no legal obligation to release the complete source code.

AGPL fixes this - it recognizes SaaS web applications too as a "distribution" of GPL software. So if a source code is licensed under AGPL, anyone who uses it to create web applications and makes it available to the public is now also legally obliged to provide the complete source code if any user requests for it. (And ofcourse, as with GPL the user is free to use the source code as they want, freedom to improve it, and even start competing SaaS services).

That is why the AGPL is currently the best GNU license to ensure that your open source code always remains open source.

It only prevents them from modifying the code and not sharing their changes and making the service available to another company.

You are still free to make private changes to AGPL code and run it for yourself (or your company).

I'm not considering "anyone" here, I'm considering copyright/license lawyers.

Would be kinda dumb for them to violate a license agreement (AGPL) while hunting for license violations.