How can even the value of g at the centre of the earth be zero and what does it has to do with the weight?
When you're at the center of the earth, you'd have equal amounts of gravitational force pulling you in all directions from the mass of the earth evenly distributed around you. Thus, the net gravitational force would be zero. This is also known as your weight. Additionally, $g = F_g / m = 0 / m = 0$.