Simple manual (if you already have TypeScript installed):
1. Download the Xrm2011.d.ts file from the link above.
2. Add it to your project.
3. In your TypeScript code reference the definition file by adding
/// <reference path="Xrm2011.d.ts" />
on top of the file.
Advanced manual (if you don't have TypeScript installed):
1. Download & install the TypeScript plugin from here.
2. Optionally download & install the Web Essentials 2012 extension for Visual Studio 2012 for improved code handling.
3. Proceed with the Simple Manual
The simple answer is that it makes coding much easier and less error prone. Please look at the following examples:
Design time type information
IntelliSense code completion
Design time check
If you see any errors or possibilities to improve the definition file, please write a comment.