Ext JS
Ext JS is a (client-side) JavaScript library created for coding Rich Internet Applications. Nowadays no one uses bare JavaScript, without an abstraction layer over the browser JavaScript implementation. Along with JQuery, Dojo, MooTools and many others, Ext JS provides JavaScript developers with a reusable toolset for modifying the DOM, building form controls, or widgets like grids, tab panels and trees.
There has been some hype lately on Ext JS being renamed to Sencha, but it was a rebranding of the company behind this product, which proposes other libraries like Raphael (SVG-related). Ext JS is still available as a standalone product on Sencha's website. Ext JS is released for free with its open source license adapt to other open source applications, but also as a commercial product, for commercial use.
Yii framework

Yii Web Programming Framework is a PHP 5 framework, mainly inspired by Prado. The goals of Yii are striving for high performance, achieved by simplicity of the design, and very loose coupling between the various components. Some of the features of Yii include the classic MVC implementation, database access infrastructure, and support for theming, caching and localization.
Yii is released under the BSD license, like
Zend Framework, so there are no limitations to its use in open source or commercial projects.
Ext4Yii

Ext4Yii is essentially a bridge between these two projects, which aims to integrate Ext JS as an extension for Yii.
Ext4Yii is implemented as a templating system, which consumes XML models where you define widget elements like buttons or handlers (in embedded JavaScript code which can take advantage of Ext-powered methods and objects).
When using the provided ExtController as the superclass of the Yii application's SiteController, the XML models are readily integrated and instantiated as Ext JS objects, without the need to include lots of .js files and event hooks.
In the Ext4Yii documentation, there are various examples of bootstrapping Ext JS widgets using these higher-level models, which deal with layout elements and forms. There are more advanced features included like AjaxMethods, which allows to call a registered PHP function via AJAX from the client side.
Learning a new modeling language can be complex, especially with verbose XML. Fortunately, there is a bundled plug-in for the NetBeans IDE which provides code completion for Ext4Yii. Even if you're not a fan of IDEs, they sometimes are very handy in providing a rapid solution to bootstrap new tools like Ext4Yii.
Conclusion
PHP frameworks have been one of the most powerful innovations in the PHP landscape in the last years, and bridging them together with a JavaScript solution is important for being able to provide RIA behavior out of the box. There are similar successful solutions available on the web, like sfJqueryPlugin or Zend_Dojo, which prove that the concept of leveraging a JavaScript library into a PHP framework is an integration worth considering.