Software was blocked from opening because it is not from an identified developer
System Preferences → Security and Privacy → Allow apps downloaded from… and select either Anywhere or (preferably) Mac App Store and identified developers (more secure option).

If you don't set it to Anywhere, to run the app right-click (⌃ ctrl-click) on the app and select Open, then take it from there (this is only needed the first time you launch the app).

I downloaded LibreOffice 4.1.2 on OS X 10.9 and tested this (I have my security setting at Mac App Store and identified developers) and was able to launch without any problems as above.