You need to be running the Node.js Meteor application. The client relies on DDP rather than REST and right now Meteor is the only server framework using DDP. There is, however, an isomorphic HTTP library which makes it easy to communicate with third party REST APIs on both the client and server. It is also easy to create REST endpoints for your Meteor collections, so that third party services which don't support DDP can communicate with your app.

It is very possible to use other client side frameworks such as Angular or Polymer, rather than just using Meteor's Blaze UI engine. However I find that Blaze's Template<--Helper<--Data<--Event system is extremely easy to understand and reason about.

Edit: Actually, they do seem to have a decoupled version of the Blaze UI that you could use just for your reactive DOM code and hook into a REST API. You just won't get any of the server-dependent features like full-stack automated data synchronization, hot-reload, live CSS injection, or Meteor's build chain. You just get a simple reactive template/helper/events system. http://meteor.github.io/blaze/