Starting with Chrome Version 42, a number of plugins, including Silverlight, stopped working.
To make EPPI-Reviewer work on Chrome for Windows, it is now necessary to explicitly enable NPAPI plugins, here is how:

1.  Within Chrome, navigate to the following URL: "chrome://flags/#enable-npapi"
2. On the resulting page, click "Enable" on the (top and highlighted) "Enable NPAPI" option.
3. Chrome should respond by telling you that "Your changes will take effect the next time you relaunch Google Chrome". Click "Relaunch Now" (just closing the current window won't be enough).
4. Go back to the main EPPI-Reviewer program. If Silverlight is already installed, everything should work. Otherwise please follow the in-page instructions.

Note: this workaround is temporary. If Google will follow the expected roadmap, Silverlight will stop working on all (current) versions of Chrome on September 2015.