In your code you need to change return `${movieName}`; to return `${this._movieName}`;

More info

Your example uses the public field declaration syntax of classes to declare two public instance properties named _movieName and _category.

Using underscores as property name prefixes is an old-fashioned (although still common) way to indicate that a property is intended to be private (although this approach offers no enforcement whatsoever).

JavaScript now has private class fields. These can be declared using the # prefix ("hash name syntax") and cannot be accessed in any way by anything outside of the class body itself.

Private fields are added before the constructor runs and are not inherited.

class Movie {
    #movieName;

    constructor(movieName) {
        this.#movieName = movieName;
    }

    showMovieName() {
        return `${this.#movieName}`;
    }
}

const movie1 = new Movie('Avengers');

console.log(movie1.showMovieName());

Update: See others answer, this is outdated.

Short answer, no, there is no native support for private properties with ES6 classes.

But you could mimic that behaviour by not attaching the new properties to the object, but keeping them inside a class constructor, and use getters and setters to reach the hidden properties. Note that the getters and setters gets redefine on each new instance of the class.

ES6

class Person {
    constructor(name) {
        var _name = name
        this.setName = function(name) { _name = name; }
        this.getName = function() { return _name; }
    }
}

ES5

function Person(name) {
    var _name = name
    this.setName = function(name) { _name = name; }
    this.getName = function() { return _name; }
}

Update: With ES6, there is a better way:

Long story short, you can use the new Symbol to create private fields.
Here's a great description: https://curiosity-driven.org/private-properties-in-javascript

Example:

var Person = (function() {
    // Only Person can access nameSymbol
    var nameSymbol = Symbol('name');

    function Person(name) {
        this[nameSymbol] = name;
    }

    Person.prototype.getName = function() {
        return this[nameSymbol];
    };

    return Person;
}());

For all modern browsers with ES5:

You can use just Closures

The simplest way to construct objects is to avoid prototypal inheritance altogether. Just define the private variables and public functions within the closure, and all public methods will have private access to the variables.

Or you can use just Prototypes

In JavaScript, prototypal inheritance is primarily an optimization. It allows multiple instances to share prototype methods, rather than each instance having its own methods.
The drawback is that this is the only thing that's different each time a prototypal function is called.
Therefore, any private fields must be accessible through this, which means they're going to be public. So we just stick to naming conventions for _private fields.

Don't bother mixing Closures with Prototypes

I think you shouldn't mix closure variables with prototype methods. You should use one or the other.

When you use a closure to access a private variable, prototype methods cannot access the variable. So, you have to expose the closure onto this, which means that you're exposing it publicly one way or another. There's very little to gain with this approach.

Which do I choose?

For really simple objects, just use a plain object with closures.

If you need prototypal inheritance -- for inheritance, performance, etc. -- then stick with the "_private" naming convention, and don't bother with closures.

I don't understand why JS developers try SO hard to make fields truly private.