I just watched your video. I looks to me like safari is having some sort of trouble determining that file is not HTML or something viewable in the browser window. It's literally loading the code of the DMG file and attempting to display it to screen, but the codec gets an error and crashes.
To confirm this, after you click the download link, does Activity Monitor show network activity that would signify a download, or is it stopped (which would signify a time-out)?
I hate to say it, but if the right-click - save as../option-click method does not work to download the files to disk, your next step would be to completely reset safari from the safari menu.
P.S. -- Do your downloads typically go directly to Applications folder? Downloads should go to the downloads folder or desktop before being opened, and shouldn't be downloaded directly to the Applications folder.