Use PHP to open the URL. Use file() or file_get_contents(), depending if you prefer to search through an array or a large string. This will give you the source of the page. Next, search for the string <input id="embed_code" name="embed_code" type="text" value="
Save the rest of the line after that as a new variable. Search that new variable for a double quote (ie, a ") and strip that out and everything after it.
Now all you need to do is to use htmlspecialchars_decode() on that string, and echo out the result.
EDIT: if you do this correctly, that final string that you echo should look like the following (but obviously with a different video and so on!)
HTML Code:
<object width="425" height="344">
<param name="movie" value="http://www.youtube.com/v/5Uk24lC3BUo&hl=en&fs=1"></param>
<param name="allowFullScreen" value="true"></param>
<param name="allowscriptaccess" value="always"></param>
<embed src="http://www.youtube.com/v/5Uk24lC3BUo&hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344">
</embed></object>
NOTE: I added in line breaks in the example code above. The actual result will all be on one line, with no whitespace.