I'll explain them in reverse because it works well that way.
Gravity is providing you with the force that you are trying to balance with centripetal force. You are trying to keep the hanging mass still. When it is still then F<sub>c</sub> = F<sub>g</sub>. Since F<sub>g</sub> is a constant (for a given hanging mass) then every time you balance the same hanging mass you are creating the same centripetal force... this is where the graph comes into it.
If you look at your graph for, say 100g, you will notice that as v<sup>2</sup> increases, so does r. F<sub>c</sub>=mv<sup>2</sup>/r. The mass of the orbiting bob is a constant and so it does not come into play. You will notice from the equation that if you increase the radius then in order to create the same centripetal force v<sup>2</sup> must increase. The slope then represents the centripetal acceleration required to generate a centripetal force to balance the downward force of gravity (remembering F=ma and that a<sub>c</sub>= v<sup>2</sup>/r).
I hope I explained that clearly enough.
EDIT: Bugger, I can't make the sub/superscripts work... are they turned off in the physics forum?
