What is TypeScript
TypeScript is a language that can be considered a superset of JavaScript. It provides types and casting to JavaScript which make writing JavaScript similar to writing C# (including providing advanced intellisense). The TypeScript code is translated to JavaScript by the IDE. You can read more about it in the typescriptlang.org site.
TypeScript infers the object types and in order to work with custom objects (like the client-side objects of the Telerik controls), it needs to have definition files for them. They are similar to vsdoc files and markdown documentation.
You can learn more about using TypeScript for the Telerik® UI for ASP.NET AJAX suite in the following help articles: