You can self-certify absence from work through sickness for seven days in the UK. Beyond that:
Can I ask if anyone has had any experience with this - what should I tell my GP?
The process is pretty much exactly the same as if you had to take time off for a physical illness.
Tell your GP the exact symptoms you've been having, and that because of these symptoms, you don't feel that you're able to work. Expect to answer questions around what if anything triggers your anxiety, and anything that you're doing to try to mitigate it at the moment. After their assessment they'll likely discuss options with you to try to treat or manage the condition (whether that's through medication, counselling or some other means), and they'll almost certainly want to schedule a follow up appointment.
Assuming they agree with your assessment that you're not fit to work at the moment, they'll give you a form that says you're unable to work until a certain date - you'll need to hand that form to your employer. Your GP may schedule a follow up appointment before the date you're set to go back to work to check that you're ok, and give you the option to extend your time off if you need it.
And as a side note, in most cases while you're sick, you're entitled to statutory sick pay - this is identical for any reason you're off sick, whether that's physical or mental health issues.