It's triggered on use of the modified software. Not just any use but use in such a way that network visitors interact with the software. If you don't use the software that way, the requirement doesn't apply even if you modified it.
If you've not modified the software, interested visitors can just get the source code from the same upstream, so why would it be required of you to host it.
However the following situation could arise and I don't see how the license addresses it. You visited a site running some AGPL service which was modified by those site operators. You decide to clone your own instance using their modified code. Since you're not modifying anything yourself you're not required to host that code. Now suppose that original site shuts down and disappears. The upstream for that modified code is no longer available. Are you on the hook for hosting it now? You've not modified anything yourself.
> Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.
Right, but suppose you disappear. The third parties are using code with your modification. Do they have to host anything? They did not modify anything themselves, but there is no upstream any more for the exact code they are using.
If you've not modified the software, interested visitors can just get the source code from the same upstream, so why would it be required of you to host it.
However the following situation could arise and I don't see how the license addresses it. You visited a site running some AGPL service which was modified by those site operators. You decide to clone your own instance using their modified code. Since you're not modifying anything yourself you're not required to host that code. Now suppose that original site shuts down and disappears. The upstream for that modified code is no longer available. Are you on the hook for hosting it now? You've not modified anything yourself.