What does this property of transition metals mean?
Take the electronic configuration of copper (atomic number $29$) $\ce{[Ar] 3d^10 4s^1}$. According to the first portion of the definition, copper doesn't have incompletely filled subshells. But, copper readily gives rise to cation $\ce{Cu^{2+}}$ that has the electronic configuration $\ce{[Ar] 3d^9 4s^0}$ with incompletely filled d subshells. So, copper is considered as a transition metal.

Now, you can guess why zinc (atomic number $30$) is not considered as a transition metal.