They just want to know you're safe. You could start by saying you're staying with a friend for a week or so and see how they take that, and actually that you were thinking it would be nice for them to have some space from you now you're a bit more independent. Whatever you do, be sure to keep in regular contact with them (like every day or two) especially at first, so that they can be confident you're okay and taking care of yourself. If you think they're going to be difficult about it then only tell them when you have already moved your stuff out, but still send them regular emails at least, letting them know you are eating well and getting some work done. Answer questions like how will you pay for rent and how will it affect your studies. Behave like a responsible adult and make them feel that you have thought through what you're doing and it's safe and positive - even if they're not okay with it they will have a hard time justifying their objections to anyone else.
...Also, bird analogies are good. You're finding it hard to spread your wings, feeling couped up, sick of living on crop milk, ready to take on the bills...
