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The feather and the ball both experience a gravitational force towards the centre of the earth which scales in proportion to their mass (the force is greater on the ball than on the feather).
The force manifests itself as an acceleration which scales in inverse proportion to the mass (the same force would make the feather accelerate faster than the ball).
So the proportionalities cancel:

in tabular form-
relative mass for same force for same gravitating body
Feather : low : easy acceleration : low coupling to gravity
Ball : high : sluggish acceleration : higher coupling to gravity

in algebraic form -
acceleration = force/mass = (const. x mass) / mass

if the acceleration is the same, so will the kinematics and they will "fall at the same rate".
Note that there are actually two kinds of mass in this - inertial (that resists acceleration) and gravitational (that couples to a gravitational field). The fact that these are the same is "deep" and leads to Einstein's Genral Theory of Relativity.

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Q: Why does a feather fall at the same rate as a ball when both are dropped from the same height and vacuum?
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