Drag one file, or more (directories are NOT supported) to droplet icon on dock. Files will be put into your Dropbox public folder, auto-synched and URL to these files will be put in clipboard. When you drop file with same name as already existing file in your dropbox, filename will be appended with unix timestamp, to make the name unique.

Important:
Open 'Dropbox-droplet.app/Contents/Resources/script' file (use Show Package Contents) and change the value of DROPBOX_USER_ID = '...' to your Dropbox ID.