The version on the website is not compatible with mac 10.5.8. The version supports versions of Mac OS X from 10.6 to 10.9. However, you have the possibility to download the correct version for your Mac using the archive page.
Access the website and then consult the available table. For your version of Mac, you will need Adobe Flash Player 11.x. The latest one is not supported. Download the correct archive, unpack it and access the folder then install the correct version for your computer/laptop.