How to Add a Website Background Image to One Page?

Use Conditional Tags: http://codex.wordpress.org/Conditional_Tags
Or a custom template for homepage: http://codex.wordpress.org/Stepping_Into_Templates

Then it’s a matter of applying a style / class when, for example, is_front_page() (or sometimes) is_home() (for conditionals), or on your custom template. It’s mostly a matter of CSS then.

Update/Alternative:

Use Chrome to learn body class. Apply style accordingly:

body.home {
    background:url(images/home_background.png) no-repeat;
}