
While using this Blogger feature, you can either subscribe to a blog's feed anonymously or can make it public so that others can also see the websites you're following. It all depends on your choice.
While using this feed reader, make sure the URL you're providing to subscribe to must offer a valid RSS feed to make it work correctly. If it's not the case, you may not be able to subscribe to the website.
To access the Blogger's built-in feed reader, click the Reading List option in the dashboard menu.

Upon clicking it, you'll find the MANAGE button on the top right side. It's for managing existing blogs you're following as well as for following new blogs.
To follow a new blog, click on the Add button.

It'll open a new dialogue box prompting you to enter the website's URL you want to follow. You can enter more than one URL by clicking the Add another option.

As I said before, it'll only work if the website's URL you're providing offers RSS feed URL for the readers. If not, it won't work.
After providing the URLs of the blogs' you want to subscribe to, click the Next button.

As you can deduce from the image above, you can either follow the blogs publicly or privately. The former option will add the list on your Blogger profile, and everyone can see which blogs you're following.
And, the latter case is self-explanatory. Once completed, you can see the list of blogs you're following in your management dashboard.

You can change your following preference from private to public and vice versa through this list. The trash bin icon on the right enables you to remove the blog from your list. That's how you unfollow a blog.
I use it regularly to follow some of my favorites, and it gives me the convenience to check out their content without ever leaving my Blogger dashboard.