How to get custom favorites icons in Safari 8 on Yosemite?

This is not related to OS X, so I’ll let the moderator move this question to other network. But I’m still going to answer this. The one you’re talking about is Apple Touch Icon which can be implemented by using the markup below:

<link rel="apple-touch-icon" sizes="57x57" href="/apple-touch-icon-57x57.png">
<link rel="apple-touch-icon" sizes="114x114" href="/apple-touch-icon-114x114.png">
<link rel="apple-touch-icon" sizes="72x72" href="/apple-touch-icon-72x72.png">
<link rel="apple-touch-icon" sizes="144x144" href="/apple-touch-icon-144x144.png">
<link rel="apple-touch-icon" sizes="60x60" href="/apple-touch-icon-60x60.png">
<link rel="apple-touch-icon" sizes="120x120" href="/apple-touch-icon-120x120.png">
<link rel="apple-touch-icon" sizes="76x76" href="/apple-touch-icon-76x76.png">
<link rel="apple-touch-icon" sizes="152x152" href="/apple-touch-icon-152x152.png">

There are multiple sizes because it needs to support Retina and non-Retina display.

Source: CSS Tricks