How do I show in the home page the facets shown in the product page?
Facets are built inside blocks. Just go to the admin/structure/block and search for Facet API, click configure on the Facet you would like to display on the front page, scroll to the end of the page and you will see a section Visibility options > Pages and on the right a selection named Show this block under the following pages.

You just have to add the pages where you want the selected Facet API block to appear. For instance if you type collection/*, the Facet will appear under every collection for the user to filter.