Why do objects fall at the same speed regardless of weight?

This comes down to how gravity works. Gravity pulls on an object proportional to its mass, but inertia, the resistance to acceleration, also increases proportionally with mass. These two effects cancel out exactly, meaning a heavier object experiences more gravitational force but also requires more force to accelerate, so the net result is identical acceleration for all objects. Galileo demonstrated this and Newton formalized it, with Einstein later explaining it more deeply through general relativity, where gravity is geometry of spacetime that all objects follow equally. In practice, air resistance does cause differences, which is why a feather falls slower than a rock. In a vacuum, however, they fall at exactly the same rate, as famously demonstrated on the Moon by Apollo 15 astronauts.