The "offer" probably includes the ability to save the cloud data to the local system once you have a local version installed. The 2010 key will only work to install Office 2010. (Probably only the Office University 2010 that you had, and not some other, Standard or Professional, version.) To install Office 2016 you will need to purchase a 2016 key. If you are still a student, you may be able to get a decent discount on that, but not likely to be free, although not impossible either. Anyway, to use the 2010 key, find the 2010 disc somewhere in your old collection and reinstall it. Then you can use the "offer" to migrate the cloud data to your computer. Once you have a 2016 key, if you choose to get one, you can upgrade your Office 2010 to Office 2016.
Update
I haven't dealt with MS Office since 2010 versions, and avoid subscription-based services for my data. Consequently, most of the above about Office 2016 is way off base. However, misunderstandings of the "free" offer aside, the solution for you is the same.
Since you still have the 25-character product key from Office 2010, you can reinstall that. BTW some sources suggest that doing a "repair" option rather than an install will work, and experimentation with that is up to you.
First you need the installation media for Office 2010. If you have the old disc gathering dust somewhere, find it. If you don't have, never had, the disc, then MS will let you download a copy. Go to their download page, enter the product key you own and follow their instructions.
Now that you have installation media, either insert the disc, or go to the downloaded files, and run the installer. (The disc will probably AutoStart, but if not, find the disc in My Computer and double click it.)
Depending on what the installer detects on your computer, you will have one or more of the following options: Install Office 2010, Re-install Office 2010, or Repair Office. Choose the one that's available, that best fits your plans. FWIW I'd go for a complete new, total, installation, to be sure that all programs are installed, and nothing is 'linked' to the expired version you upgraded to.
Finally, save you install media somewhere in case you need it again; virus issues, buy a new computer, etc. Enjoy the perpetually licensed version, subscription free.
I had this issue as well where I had a Visio 2013 (.vsdx) and it would not allow editing of embedded Excel files. So I had to convert one of two ways:
1. On a machine with Office 2013 save the file as an older version on Visio (.vsd) and then open it on the Office 365 machine and it works fine. Also, you will be able to save the file back to a .vsdx file on the Office 365 machine and open and edit it normally.
It appears that once you save it as a .vsd file on 2013 environment, then open the file, edit it on 365 environments and then save back to the .vsdx file. After this, the file can be edited normally on the Office 365 machine.
It appears Office adds all the versions to the file so that it is now editable in any office environment (2013, 2016, 365). But you have to start with saving the file as .vsd on an Office 2013 machine.
The issue is that Office 365 and Office 2016 apps do not work well on the same machine.
Best Answer
Office itself will stay installed - but when you logon it will tell you that you no longer have a valid subscription. In some cases you can still open files, in others - office will close if you don't sign in with a proper user (I know Visio does this).
If you want to continue to use office you need to get another subscription or purchase a key to re-activate office. If you have installed office from say an E-3 subscription and then go to Home subscription - you will need to uninstall and re-install office (as they are different SKU's).