&& depends of the evaluation of the right expression when left one is true, || doesn't. You could rewrite it to:

if(
    $users == 'all' ||
    ($_POST['user'] == 1 && $users == 'admins') ||
    ($_POST['user'] == 0 && $users == 'mods')
)

And it'll be the same.

The operator precedence sheet you link to states as a separate note:

Although = has a lower precedence than most other operators, PHP will still allow expressions similar to the following: if (!$a = foo()), in which case the return value of foo() is put into $a.

So, in effect, an assignment inside an expression will be treated somewhat like a sub-expression. Exactly how and when this will happen isn't clear from the documentation, which just states that "similar" expressions will work this way.

As your link https://www.php.net/manual/en/language.operators.precedence.php already explains, the table is ordered by precedence.

First, you've got in line 3 the type cast (https://www.php.net/manual/en/language.types.type-juggling.php#language.types.typecasting), than the 6th line (arithmetic operators).

Your code is equivalent to

Copy$x = ((float)'100') / 100;

or

Copy$y = (float)'100';
$x = $y / 100;

respectively.

This is an assignment, not a comparison. Besides, you have a function call which is needed to the assignment. Then the order is:

1) Function call returning false;
2) Assignment of false value to $a;
3) Negation of $a as !false, i.e., true.