Its very easy.
1) Ask your host if they support ASP or PHP scripting (note its ASP not ASP.Net). If they don't support either then it won't work.
2) Make a folder on your desktop called Weatherflash.
3) Unzip the WeatherFlash files into it making sure that you keep the file structure in the zip intact.
4) If your host supports PHP then you will need to edit the site.txt file using notepad and change the ASP to PHP on the appropriate line. (its obvious which one)
5) upload the whole weatherflash folder from your PC to your web server using your ftp program. Place it in the root of your web servers html section.
6) Go to
www.yourdomain.com/weatherflash and it should load.
Then you can try placing components on the page using WeatherFlash's built in WYSIWYG interface. If there are any problems that need resolving such as parent paths on an asp server or file permissions on a php server then they will show here as the components wont stick when you lock the page.