Hi Soeren – premium themes from WordPress.com can only be used here. The download button is an error, it should not be showing. Since you only purchased the theme yesterday, you can issue yourself a refund at:
https://wordpress.com/me/purchases
You have two alternatives at this point:
1. You can repurchase Little Story for your self-hosted site at Creative Market:
https://creativemarket.com/automattic/4242-Little-Story
2. If you’d like to stay with WordPress.com, the Business plan includes all premium themes and now allows you to install plugins. You can learn the differences between WordPress.com, the WordPress.com Business plan, and self-hosted WordPress here:
https://en.support.wordpress.com/com-vs-org/