Newton's law says that the force $\vec F$ exercing on an object produces an acceleration $\vec a$ such as :
$$\vec F = m_I \vec a$$
where $m_i$ is the inertial mass of the object.
On the other side, in your experience, the force is the gravitationnal force (the weight) $\vec P$ which is
$\vec P = m_G \vec g$, where $m_G$ is the gravitational mass, and $\vec g$ is the gravity acceleration.
The equivalence principle says that the inertial mass and the gravitational mass are equal, so $m_G = m_I$.
You have $\vec F =\vec P$, that is $m_G \vec g = m_I \vec a$
But $m_G = m_I$, so the acceleration is $\vec a = \vec g$, and this does not depends on the mass.