Problem installing Box Sync 4 application on El Capitan
Short story: drag and drop the Box Sync.app from the DMG file to your /Applications folder and run it from there. It takes a moment to present the login prompt but it does work from there.

Long short I tried the Box installer on my Mac and was getting the same issue as yourself. Watching in Activity Monitor, after I double-click on the Box Synx.app within the mounted DMG file it prompted for my password and then seconds later disappeared from Activity Monitor with no explanation.

I ran fs_usage to watch what the installer was doing and it showed that it was writing to a log file in /Users/MYUSERNAME/Library/Application Support/Box/Box Sync/Box Sync-4.0.6974.log. Each time I tried to run the installer it wrote the following lines to the log and exits.

[36m2015-12-09 10:54:09.646 72816 INFO    MainThread           mac_self_updater      Mac self updater. user: 501, elevated user: 501, running from: /Volumes/Box Sync Installer/Box Sync.app[0m
[36m2015-12-09 10:54:09.646 72816 INFO    MainThread           mac_self_updater      installing into: /Applications/Box Sync.app[0m
[36m2015-12-09 10:54:09.648 72816 INFO    MainThread           mac_privileged_execut _get_installed_version: /Volumes/Box Sync Installer/Box Sync.app/Contents/Resources/com.box.sync.bootstrapper --version[0m
[32m2015-12-09 10:54:09.666 72816 DEBUG   MainThread           mac_privileged_execut Executable should be installed/updated. Current version 0, bundled version 2[0m
[36m2015-12-09 10:54:09.667 72816 INFO    MainThread           mac_privileged_execut Executing command /Volumes/Box Sync Installer/Box Sync.app/Contents/Resources/com.box.sync.bootstrapper with arguments [u'--install_sync', u'--sync_app_path', u'/Volumes/Box Sync Installer/Box Sync.app', u'--install_path', u'/Applications/Box Sync.app', u'--parent_process_id', u'72816'][0m
[36m2015-12-09 10:54:09.667 72816 INFO    MainThread           mac_privileged_execut Requesting create execute authorization ref[0m
[31m2015-12-09 10:54:23.938 72816 ERROR   MainThread           mac_bootstrapper      Failed to launch new sync app.  Error message: You can’t save the file “Box Sync” because the volume “Box Sync Installer” is read only.[0m
[31m2015-12-09 10:54:23.944 72816 ERROR   MainThread           mac_sync_app_delegate Unable to launch application via the bootstrapper.  Double click installation will fail silently![0m


For some reason it is trying to change one or more of the files within the DMG (perhaps just to change permissions), but it can't get write access within the read-only DMG so it silently fails.

Basically there is a bug in their software that needs to be fixed. In the meantime you can work around it by copying the application to a place where you it can modify the files.

I've left a message on their support forum. https://community.box.com/t5/Help-Forum/Mac-installer-fails-on-double-click/m-p/8641