The no-cloning theorem states that it is not possible to have a quantum state $|\psi\rangle$ evolve into two separable (non-entangled) copies described by the tensor product state $|\psi\rangle|\psi\rangle$.
The proof boils down to the simple observation that when expressing $|\psi\rangle$ in some basis ${|0\rangle, |1\rangle, |2\rangle, ...}$:
$$|\psi\rangle = \alpha_0 |0\rangle + \alpha_1 |1\rangle + \alpha_2 |2\rangle + ... $$
the cloning operation would be a unitary evolution of the form:
$$U(\alpha_0 |0\rangle + \alpha_1 |1\rangle + ... ) = \alpha_0^2 |0\rangle|0\rangle + \alpha_0 \alpha_1 |0\rangle|1\rangle + \alpha_1 \alpha_0 |1\rangle|0\rangle + \alpha_1^2 |1\rangle|1\rangle... $$
This leads to a contradiction, as the unitary operator $U(..)$ is linear and can never create amplitudes like $\alpha_0^2$ and $\alpha_0\alpha_1$ that are quadratic functions of the $\alpha_i$.
So the linearity the author is referring to is the linearity of the unitary evolution. In quantum physics evolution is described by unitary operators which transform incoming states into outgoing states that are a linear combination of the ingoing states.