When you go to an update page from the results page it is passing the id for that record so that the recordset on the update page knows which record to select.
If you wanted to make it work without putting the id in the URL you would want to add a form around the update button and set the button to be an image type input, for the value you would bind the id from your results recordset and for the name use the same one as the URL parameter that you see. The form must have the method specified as post.
This will allow you to click on the update image and rather than just link to the update page with the id it will post to the update page with the id so you will not see it in the URL. If you do it like this though you will need to find any references to the get variable on your update page and change them to post, like this:
find all gets for the parameter that is passed in the URL
$_GET['<parameter name>']
replace this with the posted version of this element
$_POST['<parameter name>']
