Let's say you have a medical emergency, like breaking a bone. In this case, you'll almost certainly go to the hospital. With private insurance, you can go to any hospital—you are not bound to specific institutions or networks.
Your coverage will reimburse you for overnight stays and any immediate treatments, as detailed in your policy. The billing will be handled directly between the hospital and the insurer.
After you are released, you might require a few follow-up visits to a doctor and/or physiotherapist. These types of follow-up visits would be covered, as well as any necessary equipment like a wheelchair or crutches these would be covered as well according to the extent of your plan.
For the doctor visits and medical aids (e.g. crutches) you will receive a bill from to your home address. You can then submit this bill to the insurer by taking a picture and uploading it through a smartphone app to receive a refund.