The gradient is normal to the level curves. If you know that the level curves are concentric circles centred at the origin, then you know that the gradient points either toward the origin or away from the origin, because these are the only directions normal to circles centred at the origin. Its easy to see that the function increases as you move away from the origin, so the gradient must point away from the origin.

Alternatively, you can just calculate the gradient at a point p and see that it is proportional to p.