I've read the instructions on this but they're very vague and assume a lot from the reader. Can you help?

8.1. Upgrading the Joomla Core
As for Joomla itself, the upgrade can be performed only once.
What do you mean by "only once?"

If the Joomla upgrade restores a file that is patched by JMS, the installation
patch will have to be re-applied.
Can you post the steps to do this?

In some cases Joomla might restore a file that will require the installation
directory to be removed in order to access the back-end. In this case, you will
have to rename or delete the installation directory in order to Jms Multi Sites.
This is very hard to understand and I think you're missing a word or two near the end.

Perhaps you could just run us through the proper steps to upgrade...for example:

1) Download and install the latest Joomla upgrade on the master site.
2) Check this and check that.
3) Do this if something is wrong with this or that.
4) Done.

1) In Joomla, you have a menu for Joomla update.
Click on this menu and perform the update/upgrade.
When done, joomla update your files on the disk - That are shared by all the websites. So should be done only once.
It also potentially update the DB structure.
In this case, you have to go in each slave site and perform the "extension manager" / Database => And click on Fix button

2) Go in JMS Multi Sites / Check Patches => Install patches

3) In case of an upgrade (ie. J2.5 to J3.2), the /installation directory contain the old joomla version (here 2.5) to restore a correct /installation directory, you have to remove the /installation directory before doing (step #2)

That all.