Because you have a repeater inside of a repeater. A group field is a repeater that always has a single row. You cannot refer to the repeater with the name “group_main_info_phones” to update the field.
Like I said, you need to use field keys
$value = array(
// field_XXXXXXX is the field key of your nested repeater
// it's value is an array of rows for the repeater
'field_XXXXXXX' => array(
// each row of the repeater is a nested array
// that holds field key => value pairs
array(
// field_YYYYYYY is the field key of "name" field
'field_YYYYYYY' => 'Jon',
// field_ZZZZZZZ is the field key of "phone" field
'field_YYYYYYY' => '0897654007'
)
)
);
// field_AAAAAAA is the field key of the group field
update_field('field_AAAAAAA', $value, $the_record_id);
