How do I add a background image to flutter app?

Scaffold doesn't support any concept of a background image. What you can do is give the Scaffold a transparent color and put it in a Container and use the decoration property to pull in the required background image. The app bar is also transparent.

Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Welcome to Flutter',
      home: Container(
        decoration: BoxDecoration(
            image: DecorationImage(
                image: AssetImage("images/logo.png"), fit: BoxFit.cover)),
        child: Scaffold(
          backgroundColor: Colors.transparent,
          appBar: AppBar(
            elevation: 0,
            backgroundColor: Colors.transparent,
            title: Text('My App'),
            centerTitle: true,
            leading: IconButton(
                icon: Icon(
                  Icons.list,
                  color: Colors.white,
                ),
                onPressed: () {}),
          ),
        ),
      ),
    );
  }

Use BoxDecoration as the decoration attribute of the Container:

  Container(
    decoration: new BoxDecoration(
      image: new DecorationImage(
        image: new AssetImage("images/logo.png"),
        fit: BoxFit.fill,
      ),
    ),
  ),

    return MaterialApp(
      title: "MoonLight",
      home: Container(
        decoration:new BoxDecoration(
            image:  new DecorationImage(
              image: new AssetImage("graphics/moon.jpg"),
              fit: BoxFit.cover,)
        ),
        child: Scaffold(
          backgroundColor: Colors.transparent,
        ),
      ),
    ),