Maybe Microsoft disagrees, but I see TypeScript as something intended to make itself obsolete: eventually, when all browsers are running ES9 (or whichever version we decid has fixed "all" of JavaScript's problems) or WebAssembly (and we're all writing front-end code in C#/C++/Ruby/whatever), then TypeScript will just fade away, by design. For now, it's kinda like JQuery in motivation if not literally architecture: it shields you from browser inconsistencies (in this case, where you don't know what ES version the target is running) and helps you avoid JavaScript's headaches. Just as JQuery became less necessary as time went on-- not because it got worse but because browsers got better-- so too will TypeScript.
Dart arrived with much fanfare though, whereas TypeScript was more low-key, and it's taken a while to become mainstream. Google's announcement that Angular 2 would use it has brought it a lot more attention from outside of the traditional MS/.NET dev crowd.
Microsoft have also been shifting the project's focus towards a more Babel-esque one, emphasising greater interoperability with existing (untyped) JS libraries, and support for new ES6/2015 features (and beyond).
1. https://channel9.msdn.com/Shows/Going+Deep/Anders-Hejlsberg-...
That actually existed internally at MS, along with many other projects to improve developing large JS codebases, most of which have fallen by the wayside. There was even a Flow-like compiler that worked with type annotations in comments. TypeScript has won out, although there is a LOT of code written in these other projects.