What do you mean by “home page”? By default, the home page is automatically constructed out of most recent posts. In your case I think you are talking about a separate WordPress page, right?
Problem is, the [podlove-web-player] shortcode needs the context of which episode to show. Right now, it is not possible to display the player for a specific (or the most recent) episode using that shortcode.
You can however create a Podlove Template that’s brings up the player for the latest episode and insert it into the page using the [podlove-template] shortcode. If you like I can give you an example.