Screens Connect supports OS X 10.6.8 or later. To make sure you install the correct version, please download from https://edovia.com/screens-connect.
If you're having trouble installing the latest version, try rebooting your Mac first.
If you're still having trouble, and you're running OS X Yosemite or earlier, try repairing disk permissions. (Disk permissions are managed automatically by the operating system in El Capitan and later.)