In short, the solution was to "unblock" the executable after downloading it, but I couldn't find any guidance on this.
Here's what I was seeing, and how to solve it.
Google's Music Manager is used to upload music into Google Play's My Music area. I've liked this service, and use it on a number of machines, mainly running Windows 7 and Windows Vista.
So, from within Google Play, My Music, you choose Upload Music:
Which brings you to this screen, where you to the Download Music Manager screen:
When you click on the Download Music Manager button, you download an installer:
So, if you just click on this file, you will get a message that says something like "connecting to the internet". But it just stays there.
Strangely enough, I was unable to reproduce this to provide the screen shot. Even after un-installing Music Manager, and checking for new Windows Firewall holes, I couldn't reproduce the problem with installation.
To fix the issue, you open up your downloads folder by clicking on the context menu for the downloaded file, and choosing "Show in Folder".
Once in the downloads folder, right click on the musicmanagerinstall.exe file, and choose properties in the pop-up menu.
Click the Unblock button.
Click on OK, then double click the musicmanagerinstall.exe to begin the install. It should work fine now.
I think that this is a good safety feature - executable files off the internet should not have a high level of privileges when they are executed.
It appears that this installer is a very small file, but that it actually goes back out to the internet to download more files for the installation. In contrast, when I installed iTunes, it downloaded a fairly large (gigabytes) installer, which did not go back out to the internet for anything else, until it was completely installed.
Strange that I couldn't find any help on the internet on this. Not even in the Google Help area.
