Try using a different browser if you are using Firefox. Also, please send a message to our Support department regarding this issue. Thanks :D.

Probably it is connected to issue described here. I'll forward this post to our Support and they should be able to sort it out once they get here.

Pretty much what she said, however that particular post may not be too informative out of context. Using the Downloader will almost certainly fix your issue and allow you to download your games.

As for the browser download, it seems to get borked when it encounters convoluted Internet connections, but we're still gathering information about this. What kind of connection are you using? Is it a private connection (like DSL), or a university/corporate connection? Are you using proxy servers? Is your IP address static or dynamic?

If possible, please use these websites to check your IP address and let me know if the IP is identical on all of them:
http://whatismyipaddress.com/
http://www.whatsmyip.org/
http://www.whatismyip.com/
http://www.whatsmyip.us/
http://whatsmyip.net/
http://www.whatismyip.org/
http://www.ipchicken.com/

The idea is that I want to check if your network assigns you a different IP when you connect to different servers across the globe. With any luck, these websites are somewhat spread out around the world.

All you need to do is install the Downloader, then go to your My Account page. Once there, make sure the "GOG Downloader mode" is switched to ON, and click the appropriate installer button.
When you first use the Downloader, your browser will probably ask if you want to use it when clicking that kind of button (or link). Confirm, and wait for the Downloader to launch and start the download automatically.
In case you experience any issues, please let me know.