1. Analyze Statement 1

   Statement 1 says: An astronaut in an orbiting space station above the Earth experiences weightlessness.

   This is true.

   • The astronaut is still under Earth's gravitational force.
   • But both the astronaut and the space station are falling together toward the Earth with the same acceleration.
   • Hence, there is no normal reaction force exerted by the floor on the astronaut.
   • Since apparent weight is the normal reaction, the astronaut feels weightless.

2. Analyze Statement 2

   Statement 2 says: An object moving around the Earth under the influence of Earth's gravitational force is in a state of free-fall.

   This is also true.

   • In orbital motion, gravity alone provides the centripetal force.
   • So the object is continuously falling toward the Earth while also moving tangentially.
   • Therefore, an orbiting object is indeed in free-fall.

3. Check whether Statement 2 explains Statement 1

   Yes, it does.

   • Because the astronaut and the space station are in free-fall together, no support force is needed to hold the astronaut against the floor.
   • Thus the astronaut experiences apparent weightlessness.

4. Evaluate options

   • A: Both statements are true, and Statement 2 correctly explains Statement 1. ✅
   • B: Explanation part is wrong. ❌
   • C: Statement 2 is not false. ❌
   • D: Statement 1 is not false. ❌

Therefore, the correct option is:

$\boxed{A}$