Can an abstract class contain a private method and why?

Yes, maybe.

A private method in an abstract class is visible and accessible only within the abstract class itself. This can be useful for implementing internal logic necessary for the operation of other methods of this class (for example, auxiliary operations), while hiding this logic from derived classes.

Methods that should be available for overriding or use in child classes are declared as protected or public.

<?php

abstract class AbstractExample {
    abstract public function doSomething();

    private function internalHelper() {
        // Private method, visible only within AbstractExample
        echo "Performing internal operation.\n";
    }

    protected function protectedHelper() {
        // Protected method, also visible in derived classes
        $this->internalHelper(); // Calling private method from within is allowed
        echo "Performing protected operation.\n";
    }
}

class ConcreteExample extends AbstractExample {
    public function doSomething() {
        echo "Doing something specific.\n";
        $this->protectedHelper(); // Access to protected method
        // $this->internalHelper(); // Error: No access to private method
    }
}

$obj = new ConcreteExample();
$obj->doSomething();