Several of my featured products have product options, why do these options only appear on the product details page?

In order for the Product Options to be displayed for Home-Page Featured and Category Featured items, the setting Allow 1-Click Ordering of Category & Home-Page Featured Products in the Settings > General Settings page must be set to Yes.

 
This setting is available in the Display Settings table in this page.

Note: Featured products that have Product Options cannot be added directly to the cart by clicking the Buy button, which is why you are taken to the product page when you click on the it. However, featured products without Product Options can be added directly to the cart from the Home Page.