First, remember that the Category Icons plugin just generates images tag, nothing more. Thus, you can first work with an image in your theme, and when you are satisfied, replace that image with the get_cat_icon tag.
Through the Template Tags panel, your template files are scanned and patched if they’re writable. If not, informations about where you should paste get_cat_icon() in your files are displayed : line number and the column number. These locations are just where you COULD paste the tag, not where you MUST paste it. It’s up to you to find the appropriate location within your template files. The tag is usually within the Loop. A blue arrow shows you where to paste the followin code :
<?php if (function_exists('get_cat_icon')) get_cat_icon(); ?>
Nota Bene : If you don’t want your files to be patched automatically, make sure it’s not possible to write into them by checking the permissions.
If you want to place it in another place than in the Loop, you should read this post.