Facebook Messenger installs the same way as Google Chrome browser - through connecting to the Internet after you click on the *.exe file and downloading-installing the needed files.
Upon double-click of the FacebookMessenger.exe file, I had my Security Software pop-up and ask whether I let the program connect to the internet. Then a window "Initializing dowload" appeared and the app got downloaded and installed on my PC (Windows 7 x64 Ultimate).
The fact that the "Initializing download" window does not appear after you double-click the *.exe could mean that your firewall automatically prevents the program from connecting to the Internet and downloading the needed installation files.
Make sure you manually let the *.exe file through your firewall. If you don't know what firewall you have or you have the default Windows Firewall active, read this: Allow a program to communicate through Windows Firewall