User expectations for web applications have increased exponentially the past few years. Users now expect applications to respond quickly to their interactions and to render appropriately for different size devices. In addition, users have pushed back against using browser plug-ins, such as Flash and Silverlight.
Some frameworks are very specific, such as jQuery which eases the process of selecting and acting on the DOM elements of a browser; and MustacheJS, which provides automatic data binding. And some are very general frameworks, such as Knockout, Ember, Angular, and React, that provide complex functionality for most aspects of your application and allow you to build custom modules of your own.
Of course, the frameworks themselves add overhead - both in terms of learning time for the developer and download time for the user. For very simple pages, this overhead might not be worthwhile; but for even moderately complex applications, a framework can manage said complexity, making your code easier to maintain, deploy, debug, and test; and freeing you up to focus less on the application plumbing and more on the code that is unique to your application.
Does it meet the needs of my application
Do you need a do-everything framework or just data binding. Is the User Interface the most important thing or is synchronizing with backend data more important. Each framework has its strengths. Determine what you need; then find the framework that suits you.
How difficult is it to learning
Look for a framework with good documentation and tutorials. Often, ease of learning is a function of your current knowledge. If you are already familiar with the Model-View-Controller pattern, it may make sense to use a framework that implements this pattern.
How popular is it?
This may strike you as a frivolous criterion, but a popular framework will have more people blogging about it; more people answering forum questions; and bugs will get found and fixed more quickly.
Will it be popular next year?
Future popularity is difficult to predict; but it may be more important than current technology. You are likely to keep this framework for a long time - possibly the life of your application and you want your technologies to remain relevant and supported.
Whichever framework you choose, you will learn it best by diving in and beginning your project.