To call a function you need to break out of the double quotes like so,
PHP Code:
$db->query_read("
INSERT INTO picks_groups
(group_title)
VALUES (
'" . $db->escape_string($event_name) . "'
)
");
Used " to close the string, and then . (concat operator) to join it with the next section which calls the escape function, and then . and " again to go back to double quoted string mode.
Another (easier to read / write for some, but not necessarily better) method is this:
PHP Code:
$escaped = $db->escape_string($event_name);
$db->query_read("
INSERT INTO picks_groups
(group_title)
VALUES (
'$escaped'
)
");