*.d.ts files are used to provide typescript type information about a module that’s written in JavaScript, for example, underscore / lodash / aws-sdk.
This will allow you to use the javascript modules without the need to convert them to ts without getting any type of error on your code.
for example, if you have a folder myAwesomeLib, with index.js and index.d.ts files
on your code, you will be able to import the code with
import { SomeMethod } from './myAwesomeLib';
or
import { SomeMethod } from './myAwesomeLib/index';
your typescript will rely on the .d.ts file to find the correct types for the SomeMethod
Edit: More about Declaration files https://basarat.gitbooks.io/typescript/docs/types/ambient/d.ts.html