I am not sure what the title should be, but the code should explain it better:
class Group {
    private $number = 20;
    public function __toString() {
        return "$this->number";
    }
}
$number = new Group();
echo $number, PHP_EOL;
echo ++ $number, PHP_EOL;
echo PHP_EOL;
$number = "20";
echo $number, PHP_EOL;
echo ++ $number, PHP_EOL;
echo PHP_EOL;
$number = 20;
echo $number, PHP_EOL;
echo ++ $number, PHP_EOL;
Output:
20
20              <--- Expected 21
20
21
20
21
Any idea why I got 20 instead of 21? Even then the code below works: 
$i = null ;
echo ++$i ; // output 1
I know Group is an object that implements __toString , i expected ++ to work with the string from __toString or at least throw an error 
The order in which the operations happen is important:
The variable will be fetched as an object, it won't be casted to an integer (or something else).
This ++ operator increments the lval (the long value) of the zval, but does normally nothing else. The object pointer remains the same. The internal (fast_)increment_function will be called with the zval which has a pointer to the object, which checks for the type first. If it's an object, it does nothing. So when your zval is an object, it is as useful as a no-operation. This won't output any warning.
Only then the echo instruction performs a string cast on his arguments: The __toString method is called and returns 20.
20 will be output.
To answer you question with a little bit of code.
$number = new Group();
echo gettype($number);
$number = "20";
echo gettype($number);
$number = 20;
echo gettype($number);
Will result in
object
string
integer
The three cases:
__toString method will called very late, when the acutal output will computed, after you unsuccessfully tried to do an math operation with it.Bonus:
This will work:
$number = new Group();
echo 1 + "$number"; // 21
It converts you object into a string, which could be converted into a number for a math operation.
I think it might be clearer with just changing the names of the variables like this :
class Group {
    private $number = 20;
    public function __toString() {
        return "$this->number";
    }
}
$group = new Group();
echo $group;//print 20 as per your __toString function
++ $group;
Now it seems obvious : what is supposed to do a '++' operator on a object of type group ??
Why don't you just:
class Group {
    private $number = 0;
    public function __construct($number = 0){
        $this->number = intval($number);
    }
    public function __toString() {
        return number_format(++$this->number); // pre-increment
    }
}
$g = new Group();
echo $g; // 1
echo $g; // 2
I use something like this to format offsets in tables.
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