Yeah in order to use delayed_job you need to be paying for one worker process. The worker process will only process your delayed_jobs from the queue, and then your normal web process (they give you one for free) will process all of your web requests. You scale web and worker processes separately.
A heroku ps
will show you which processes you have running and you should have at least 1 web and at least 1 worker in order to process web requests and delayed job processes.
You can either use the command line or the web interface to add processes, command line to add your worker would be:
heroku ps:scale worker=1