Please 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.
If after trying this, the Library Manager still does not open at all please get in touch with us at spitfireaudio.com/support and we will troubleshoot the issue further with you.