Edit: devonbarrett seems to have removed the DO NOT USE from the readme since earlier today. If you are still having issue you should post an issue.
I expect your issue has to do with the current way Meteor renders templates. While not actually true, meteor does not support updates to the DOM via anything other then handlebars. This will be fixed by early next year if not before then. See Meteor.UI for details on how things will change.
Also you should know that Dropzonejs package does not work just yet, see the DO NO USE at the top of the readme. I was talking with the person who is making that package, via IRC #meteor, and ze is trying to make it work with both current meteor and with the new rendering engine (Meteor.ui).
You have a few options:
Wait for Dropzonejs to be usable. I am in line with you :)
Use the new template engine at the risk of bugs.
use something other then Dropzonejs, for example impicker. Note that I have not tried it.
Finally if you are new to meteor you really should checkout EventedMind. That site is full of very useful videos that can help you get an idea of how meteor works and the options you have when working with it. There is also a few videos on file upload.
I didnt try DropzoneJS with meteor but i think you should try using meteor constant regions.
Constant regions allow non-Meteor content to be embedded in a Meteor template. Many third-party widgets create and manage their own DOM nodes programmatically. Typically, you put an empty element in your template, which the widget or library will then populate with children.