The key word is "experiences". They have gravitational mass in a gravitational field (meaning they have weight), but the sensation of is weightlessness due to their accelerated reference frame.
Apparent weight is a concept that takes into account an accelerated reference frame. For example, if you are in an elevator that accelerates up or down, the amount of force exchange between your feet and the floor changes as the motion of the elevator changes. It is reduced if you are accelerating down and increased if you are accelerating up. For the case of weightlessness, the elevator would accelerate downwards at the rate of g. So there would be no need for a contact force with the ground. One would float in the elevator and feel weightless..
Now regarding a satellite, the capsule in a stable orbit is always accelerating towards the planet at a rate of g (the gravitational field at its location), so one inside the capsule would also need no contact with the "floor" and would feel weightless.
Because there are no external contact forces pushing or pulling on the astronaut.
There are a few concepts that need to be covered to understand this question. The first is weight. What is it? For an in-depth answer, check out this page, but basically, weight is the force of gravity acting on your mass. That is why you weigh less on the Moon then you do on Earth. Mass is different from weight, since mass isn't dependent on gravity.
Now that you understand what weight is, weightlessness is relatively simple: The absence of the perception of weight. Note that this doesn't mean gravity is non-existent, far from it. To explain weightlessness further, we must first know about contact and non-contact forces.
A contact force is any force which acts upon an object when in contact with that object. For instance, when someone pushes you, a contact force is being acted upon you. Or, when you throw a ball, you apply a contact force to that ball. You are being acted upon by a contact force even when you sit down, as your chair balances the force of gravity with its own upward force, commonly called the normal force.
A non-contact force is any force that acts on an object without being in contact with that object. This includes gravity, which acts on you even when you jump and stop contact with the Earth, and electro-magnetism, which causes magnets to connect with each other without needing to be in contact.
What do these forces have to do with weightlessness? Well, the fact is that weightlessness happens when all contact forces are removed. For example, take skydiving. After you jump out the plane, and before air-resistance starts to properly kick in, you feel weightless. This is because only gravity is working on you, a non-contact force. This is usually called free-fall. Take note that your weight doesn't actually disappear, rather your perception of having weight disappears, since the force of gravity, or weight, can't be felt without any other opposing forces.
How does this relate to astronauts? Well, astronauts orbiting Earth are in a constant state of free-fall towards Earth, and as we've discussed, free-fall equals only gravity acting on an object. Since the pod and astronaut are orbiting at the same speeds, there is no contact force, thus the astronauts experience weightlessness.
I hope I helped!