It is a characteristic of most browsers that if they understand a file, they will display it rather than serve it as a download. You can then use the browser's "save" dialog to save the file to your computer. This is the case with exported CSV files from simpleContact Pro.
If you would prefer clicking the export link to trigger a download, you can do this:
Open sc_admin/export.php
Delete line 29 and replace it with these lines:
header('Content-Type: application/octet-stream');
header("Content-Transfer-Encoding: Binary");
header("Content-disposition: attachment; filename=\"".$file."\"");
readfile($file);
exit;
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Please note: Different browsers will handle it in different ways (some will give you a "save" dialog, whereas others will automatically save the file). These inconsistencies make me uncertain whether this approach is preferable in all cases, but by all means give it a try!