A piece of the headphones is stuck inside

You can try to pry it out with tweezers. With my standard pair of tweezers (random chemist) i have successfully removed various parts of debris, solving the problem. Sometimes displacing the foreign object with a gentle nudge can be enough to remove said object. If there is any resistance at any point, your tweezers are probably too large and you should abort mission. I should have stated that you should switch your phone off and that you should take extreme care not to damage the component or push the object in any further. If this does not solve your problem you can use the guide on ifixit to replace the part.