So, first up; this is a pretty short guide, I think, I hope, lol, I'm going to try not to be too wordy and dad-joke-y this time. We'll see if I manage.
We'll need a mod that'll error on us; I presume if you're here, you do have one! The one that absolutely will error on 2.0.15 without fail is SubAccounts, so we all need to know how to fix that at some point. I have yet to run into a mod that errors on SMF 2.1, but the principle is the same, so, follow along here, too. When the package installer encounters errors, it will highlight what steps of the installation process went awry in its test installation. These are marked red in the steps at the bottom of the installation page.
This tells us exactly what to do, incidentally.
Disclaimer: Take backups of your files before you do anything.
Disclaimer Two: If you're not that good with PHP, do it ONE THING AT A TIME. You are very likely to implode it at some point, you'll need to be able to just undo it easily to get it working again and then stare at it to figure out why it imploded.
So, find all the steps that failed, and click the little page icon to the far left side of each step. Copy and paste what it gives you on the pop-up window. It tells you, at the top of that window, what to find in the file it errored in, and at the bottom, what to do with it (add before, add after, or replace).
Then, we're going to open the source or template file that it failed on, and search for a bit of the first part that came up in the window we clicked to get the codes. You may need to change up how much you search for to find what you're looking for, but most of the time, you'll find it okay the first time. These automatic installations fail because of other mods, updates, or admins, changing the files and altering what it's looking for just enough that it can't find it anymore. Thereby, finding it requires a human eye, as you'll need to be able to discern that the line you're looking for, is the line you're looking for, even with its changes.
Once you're SURE you found it, add in the second code block from the pop-up, either before, after, or replacing the original bit (it will tell you which to do in the error row on the installation page, so make note of it).
For those replacement lines, be sure that you keep whatever alterations were made to it before, as well. So compare the two blocks to one another, what's there and what the mod you're installing wants there, and ensure that what you end up with has both the mod that altered it before, and what the new mod wants there, too. Often, a security update to SMF changed one of the lines the package installer is looking for, and it's important to be sure to have the security update still there.
Rinse and repeat with all the steps that errored. Voila. Enjoy your new mod!
Quick Note for "This mod is not compatible with your version of SMF":
Use the emulator at the bottom of the main package manager to emulate the version of SMF that the given mod wants (you may need to get into the package files and find package_info.xml, and look for versioning numbers; you can do this through the package manager's "list files" option). It will then install just fine; this is actually caused by mod writers defining too tightly what version of SMF their mod works for (i.e. specifically defining SMF 2.0.10, instead of SMF 2.0.0 - 2.0.99, which will make it work for 2.0 for like, ever). It's not an automatic condemnation of the mod on your version, it's okay. Emulate and try anyway.