When going around a roundabout, why do I feel a force pulling me outwards?

The force pulling you outwards is what is called fictional force. Actually, there is a force pulling the car inwards and since Newton's Laws mean that you want to keep moving forwards rather than going around the bend, it feels as though you are being thrown outwards. You might then ask what is causing the inward force that acts on the car. There must be a force to make the velocity of the car change direction (around the curve). This is called the centripetal force and it is key to circular motion. In the case of the car, this force is provided by the friction between the car tires and the road. But if you swung a pendulum around your head, it would be provided by the tension in the string.