In most cases, this is due to the Spitfire App being blocked by a firewall or anti virus. Adding the Spitfire App to the exception list in the security application should allow the app to open.
If adding the app to the exception list doesn't work, try downloading and running the software available at this link. In the majority of cases you will need to run vc_redist.x86.exe though if this doesn't work, try the x64 version too.
Sometimes Windows lacks the required dependancies to properly launch Library manager. Fortunately Windows provides an update for this situation, this update is the latest in a cumulative series of feature additions and bug fixes for Microsoft Visual C++ 2015 Redistributable.
Finally, if the above advice does not help, right click on the Spitfire App .exe file, go to Properties > Compatibility > Run this Program, and select the version of Windows you are using.
If after trying this, the Library Manager still does not open at all please get in touch with us at https://spitfireaudio.zendesk.com/ and we will troubleshoot the issue further with you.