Here’s the rewritten function:

function user_the_categories() {
    // get all categories for this post
    global $cats;
    $cats = get_the_category();
    // get all of the category names from $cats
    global $catnames;
    // iterate through each category in $cats
    for ($i = 0; $i < count($cats); $i++) {
        // get the catname for this category
        $catname = $cats[$i]->cat_name;
        // add the catname to the $catnames array as the key AND value
        $catnames[$catname] = $catname;
    }
    // remove any unwanted categories from $catnames by using unset
    unset($catnames["Blog"]);
    // rebuild the array after using unset (important)
    $catnames = array_values($catnames);
    // iterate through the remaining categories in $catnames
    for ($i = 0; $i < count($catnames); $i++) {
        // echo this category
        echo $catnames[$i];
        // if there are more categories remaining, echo a comma separator
        if ($i + 1 != count($catnames)) {echo ', ';}
    }
}

Laura