array_reduce

To make it clearer about what the two parameters of the callback are for, and what "reduce to a single value" actually means (using associative and commutative operators as examples may obscure this).

The first parameter to the callback is an accumulator where the result-in-progress is effectively assembled. If you supply an $initial value the accumulator starts out with that value, otherwise it starts out null.
The second parameter is where each value of the array is passed during each step of the reduction.
The return value of the callback becomes the new value of the accumulator. When the array is exhausted, array_reduce() returns accumulated value.

If you carried out the reduction by hand, you'd get something like the following lines, every one of which therefore producing the same result:
<?php
array_reduce(array(1,2,3,4), 'f',         99             );
array_reduce(array(2,3,4),   'f',       f(99,1)          );
array_reduce(array(3,4),     'f',     f(f(99,1),2)       );
array_reduce(array(4),       'f',   f(f(f(99,1),2),3)    );
array_reduce(array(),        'f', f(f(f(f(99,1),2),3),4) );
f(f(f(f(99,1),2),3),4)
?>

If you made function f($v,$w){return "f($v,$w)";} the last line would be the literal result.

A PHP implementation might therefore look something like this (less details like error checking and so on):
<?php
function array_reduce($array, $callback, $initial=null)
{
    $acc = $initial;
    foreach($array as $a)
        $acc = $callback($acc, $a);
    return $acc;
}
?>

So, if you were wondering how to use this where key and value are passed in to the function. I've had success with the following (this example generates formatted html attributes from an associative array of attribute => value pairs):

<?php

    // Attribute List
    $attribs = [
        'name' => 'first_name',
        'value' => 'Edward'
    ];

    // Attribute string formatted for use inside HTML element
    $formatted_attribs = array_reduce(
        array_keys($attribs),                       // We pass in the array_keys instead of the array here
        function ($carry, $key) use ($attribs) {    // ... then we 'use' the actual array here
            return $carry . ' ' . $key . '="' . htmlspecialchars( $attribs[$key] ) . '"';
        },
        ''
    );

echo $formatted_attribs;

?>

This will output:
 name="first_name" value="Edward"

Sometimes we need to go through an array and group the indexes so that it is easier and easier to extract them in the iteration.

<?php

$people = [
    ['id' => 1, 'name' => 'Hayley'],
    ['id' => 2, 'name' => 'Jack', 'dad' => 1],
    ['id' => 3, 'name' => 'Linus', 'dad'=> 4],
    ['id' => 4, 'name' => 'Peter' ],
    ['id' => 5, 'name' => 'Tom', 'dad' => 4],
];

$family = array_reduce($people, function($accumulator, $item) {
    // if you don't have a dad you are probably a dad
    if (!isset($item['dad'])) {
        $id = $item['id'];
        $name = $item['name'];
        // take the children if you already have 
        $children = $accumulator[$id]['children'] ?? [];
        // add dad
        $accumulator[$id] = ['id' => $id, 'name' => $name,'children' => $children];
        return $accumulator;
    }

    // add a new dad if you haven't already 
    $dad = $item['dad'];
    if (!isset($accumulator[$dad])) {
        // how did you find the dad will first add only with children 
        $accumulator[$dad] = ['children' => [$item]];
        return $accumulator;
    }

    //  add a son to his dad who has already been added
    //  by the first or second conditional "if"
    
    $accumulator[$dad]['children'][] = $item;
    return $accumulator;
}, []);

var_export(array_values($family));

?>

OUTPUT

array (
  0 =>
  array (
    'id' => 1,
    'name' => 'Hayley',
    'children' =>
    array (
      0 =>
      array (
        'id' => 2,
        'name' => 'Jack',
        'dad' => 1,
      ),
    ),
  ),
  1 =>
  array (
    'id' => 4,
    'name' => 'Peter',
    'children' =>
    array (
      0 =>
      array (
        'id' => 3,
        'name' => 'Linus',
        'dad' => 4,
      ),
      1 =>
      array (
        'id' => 5,
        'name' => 'Tom',
        'dad' => 4,
      ),
    ),
  ),
)

<?php
$array = [
  [
    "menu_id" => "1",
    "menu_name" => "Clients",
    "submenu_name" => "Add",
    "submenu_link" => "clients/add"
  ],
  [
    "menu_id" => "1",
    "menu_name" => "Clients",
    "submenu_name" => "List",
    "submenu_link" => "clients"
  ],
  [
    "menu_id" => "2",
    "menu_name" => "Products",
    "submenu_name" => "List",
    "submenu_link" => "products"
  ],
];

//Grouping submenus to their menus

$menu =  array_reduce($array, function($accumulator, $item){
  $index = $item['menu_name'];

  if (!isset($accumulator[$index])) {
    $accumulator[$index] = [
      'menu_id' => $item['menu_id'],
      'menu_name' => $item['menu_name'],
      'submenu' => []    
    ];
  }

  $accumulator[$index]['submenu'][] = [
    'submenu_name' => $item['submenu_name'],
    'submenu_link' => $item['submenu_link']
  ];

  return $accumulator