Welcome to Adobe Forums.
Are you working on a Windows Computer or a MAC ?
Did you try download Flash Player installation setup from here : http://get.adobe.com/flashplayer/
I am working with Windows 7. I have downloaded Flash Player from the site you show below, however I get a pop up window with the install_flashplayer..... etc. and once I save the file - I cannot find it on my computer. I reiterate that I am not at all proficient in navigating to find where it is saved & nothing changes in my in ability to get a video to open.
Download the Adobe Flash Player installer directly by right clicking one of the following links and selecting either "Save target as" (IE) or "save Link as"(Firefox).
When the "Save as" dialogue box opens you can select your Downloads folder as the destination. THAT's where it'll be.