When you click the Follow button on a site, any new site content will appear on your Home dashboard in the Latest tile. You won't have to go to the site to see new content.
When someone posts on the site feed, those feed posts will also appear on your home feed.
If you click on the star next to a site, person or piece of content, you have favorited it for easy navigation. You can find all your favorites by clicking on your profile image in the upper right, then going to Favorites.