You guessed right, HTTP Headers are not part of the URL.
And when you type a URL in the browser the request will be issued with standard headers. Anyway REST Apis are not meant to be consumed by typing the endpoint in the address bar of a browser.
The most common scenario is that your server consumes a third party REST Api.
To do so your server-side code forges a proper GET (/PUT/POST/DELETE) request pointing to a given endpoint (URL) setting (when needed, like your case) some headers and finally (maybe) sending some data (as typically occurrs in a POST request for example).
The code to forge the request, send it and finally get the response back depends on your server side language.
If you want to test a REST Api you may use curl tool from the command line.
curl makes a request and outputs the response to stdout (unless otherwise instructed).
In your case the test request would be issued like this:
$curl -H "Accept: application/json" 'http://localhost:8080/otp/routers/default/plan?fromPlace=52.5895,13.2836&toPlace=52.5461,13.3588&date=2017/04/04&time=12:00:00'
The H or --header directive sets a header and its value.