There are several ways I suppose. As one example, you can request them from another site and parse the response. No comments on the legalities of republishing such data though.
As an example, the NWS in the US allow you to enter airfield IDs and get the the metar for each. The MetOffice allow it too but there you have to register first, though the same mechanism would work. The URL for the NWS is:
http://weather.noaa.gov/cgi-bin/mget...cccc=XXXX+YYYY
where XXXX and YYYY are airfield identifiers. You can add several.
To get this data on to your web page you could, most simply, include a frameset with the weather info in one frame. For a nicer looking page you could periodically grab the results of the URL above (use something like "wget", available for Windows and Unix/Linux) and then parse the results (use GREP or a simple Perl script) to leave just the bits you want. Stuff some HTML around the results and there you have it.
I can't see that it's worth the effort really - why not just have link directly,
like this?