I have had a website on bravenet for a while now. Recently I purchased a domain name (not through bravenet, but through another site).

I would like my new domain and my existing bravenet site to be one and the same.

I have added my new domain as an external domain, and I have set my domain's name servers as directed by Bravehost.

I am unsure of what the next step is. When I type in my new domain, it says that the site is under construction.

Any idea what I need to do next? Thanks.

well when you have a page that says UNDER CONSTRUCTION that means you have NO index.html page uploaded yet to be recognized

You have the placeholder page and can now start building your site

I see. So is there a way of easily transferring the files from my bravenet site to my new domain?

A domain is like a custom licence plate for a car.

You need a car (web hosting) and a licence plate (domain)

You can't fit new tyres (files) to the licence plate, you fit them to the car (web hosting).