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Why do I need to place the favicon file in the site root?

The site root directory is the best place for the favicon file.

Some older browsers and most feed readers have no way of specifying the favicon file location in the document or feed markup. They always attempt to retrieve it from the site's root directory.

My browser finds the favicon automatically even without this plugin?

Most browsers look up the favicon file in the default location automatically, but some do not. They need to be told where it is located and this is what this plugin does.

Wouldn't it be easier just to edit my theme's header.php and add the favicon link there?

Up to you. Choose whatever way is easiest for you.

If you edit the theme's header.php you will need to make the same changes again if you later on update or change your theme. With this plugin you do not need to remember to do that and upgrading becomes simple.

My theme already specifies the location of the favicon?

In this case you probably do not need this plugin. It might be still best to put the favicon.ico file in your site's root directory.

I am unsure how to create the favicon.ico file?

This is well documented elsewhere. A good place to start is to understand Wikipedia article on favicon.