How to call custom category field data in theme?

If you are using the suggested method in the question you linked, then all of the category meta fields are stored in a single option in the options database table, so to get the data in your theme you need to get that option and extract the field from there, something like,

//get the current term
$term = get_term_by( 'slug', get_query_var('term'), get_query_var('taxonomy') );
//get the saved category custom fields
$fields = get_option(MY_CATEGORY_FIELDS);
if (isset($fields[$term->term_id])){
    //extract just the needed term fields
    $term_fields = $fields[$term->term_id];
    //Now $term_fields holds all of your category fields so to get a specific field:
    if (isset($term_fields['Field-Name']))echo $term_fields['Field-Name'];
}