Why is weight equal to the difference between the Gravitational force and the centripetal force? I read an explanation of the feeling of weightlessness of an astronaut that went as follows:

  
*
  
*Gravitational force provides the centripetal force
  
*So gravitational force is ‘equal’ to the centripetal force
  
*‘weight’/sensation of weight/contact force is difference between $F_G$
  and $F_C$ which is zero 
  
*So astronaut has no sense of weight... he feels weightless.
  

What concerns me is the   

contact force is difference between $F_G$
  and $F_C$

part.
Why is this? Is it because $F_C$ is provided by $F_G$ and what is left of $F_G$ is then equalized by the normal contact force? 
 A: There is a confusion there. The $F_C$ in the third bullet point is the centrifugal force, not the centripetal force. The centripetal force is provided by gravity. But the astronaut is in an accelerated frame of reference, in such a case you have to add a pseudo force, the centrifugal force, which in this case acts in a direction opposite to gravity. If gravity is the only centripetal force, then its magnitude will be equal to the centrifugal force, and both cancel each other. If, on the other hand, the spaceship is not in a free orbit, but is accelerating to have either a larger or a smaller orbital speed than it would have due to gravity, the two forces will not cancel, and the astronaut will feel a force (either towards earth or away from it, depending if the speed is slower or faster than the free orbit) that he will interpret as a weight.
A: 
I read an explanation of the feeling of weightlessness of an astronaut that went as follows: ...

*

*‘weight’/sensation of weight/contact force is difference between $F_G$ and $F_C$ ...


If that's what you read, it's blathering nonsense. Even if you replace "centripetal" with "centrifugal", it's still blathering nonsense.
It's perhaps easier to look at what an accelerometer "feels" as opposed to what a human feels. We humans are a complex mix of hard bones, soft tissues, and multiple kinds of nerves that sense tension, compression, and acceleration. An accelerometer is a much simpler device.
Place an accelerometer on a table and it registers 1 g upward. Strap an accelerometer to the leg of a bungee jumper and it will register a highly variable acceleration, at first zero g when the bungee jumper is accelerating downward, then transitioning to a large upward acceleration (eventually greater than 1 g) as the bungee cord stretches, then back to zero as the bungee cord relaxes and the jumper starts to fall again.
There's a simple explanation for this. Gravitation affects every little bit of an accelerometer almost equally and thus produces no measurable stresses or strains. On the other hand, the normal force exerted by the table or the tension produced by the connection with the bungee jumper are propagated through the accelerometer via stresses and strains.
Another way to look at it: Accelerometers sense everyone except gravity. Accelerometers are examples of small (aka local) experimental device. No local experiment can sense gravitation. This is one of many consequences of the equivalence hypothesis.
The small extent of a human body compared to the size of the Earth means that we, too, are local experimental devices. Astronauts in the Space Station feel "weightless" because the only real force acting on them is gravitation. There's no tensile force from a bungee cord, nor is there a normal force exerted by a chair, the ground, or a roller coaster. Those astronauts feel those real forces when they return to Earth.
Weight as felt by us humans, or by an accelerometer, is the net sum of all non-gravitational forces acting on the object in question. Another name for this is "apparent weight".
Yet another name is "weight". The idea that "weight" is the mass times gravitational acceleration is a bit of a fiction. In fact, general relativity texts often use the net sum of all real forces as the definition of "weight." But what about gravitation? In general relativity, gravitation is a fictitious fore.
A: Well, have you ever felt your weight while jumping from a high place or running a long distance? Didn't you feel pain on the lower end of your feet? Well, this is because we are able to feel our weight due to the normal reaction force which the ground(or any other suface) gives us. This constant force exerted on the foot gives us the senstation of having 'weight'.
In case of an astronaut, the force exterted by the astronaut on the satellite is given by (Gravitational force - centrifugal force).  Since these two are equal, the force exerted on the satellite is zero. So zero normal force is exerted by the satellite too.
[Contact force = F(c)-F(g).]
Also, in case the astronaut is in space, the same thing happens,i.e., no surface to provide the normal force.
This zero normal force (so called 'contact' force) gives the feeling of weightlessness.
