From the error message I can see in your screenshots I can tell you that this is a specific issue that occurred with the version of Installer included in Types 3.2.5.
It didn't affect all users (depends on server settings), but for those it did there is unfortunately no alternative but to manually update Types, which should then restore the normal installer functionality.
So on your plugins page delete the current version of Types.
Download the current version from toolset.com/account/downloads
On the Add new plugins page upload the zip file and activate Types.
All software has bugs, but with our last round of updates too many slipped through and last week we adopted new procedures to improve our testing to make this less likely.
In the case of the installer bug, it didn't affect all users—it depends on the server environment, making it harder to anticipate—and unfortunately such a bug cannot be fixed with an automatic update because it was automatic updates itself that was broken, therefore requiring manual intervention.