The assertion that the wavefunction has no physical meaning might mean different things in different contexts.
Wavefunction is unobservable
In one interpretation the assertion simply means that the wavefunction is unobservable. This is a scientific fact that can be proven for example by showing that the ability to observe it would allow faster-than-light communication (e.g. using arguments similar to those that rule out single-shot fidelity measurements). The impossibility to observe the wavefunction together with its central role in quantum mechanics raises some tricky questions.
Wavefunction is similar to probability distribution
Quantum mechanics can be thought of as an extension of the theory of probability where wavefunctions correspond to probability distributions. This suggests that wavefunctions may be assigned similar ontological status to probability distributions. These may be interpreted as a description of an observer's imperfect knowledge of the state of a system or sometimes as an incomplete description of the state of a system. Therefore in another view the assertion about the wavefunction lacking physical meaning refers to the fact that the wavefunction may be interpreted in a similar way. In the hidden variables research program the respective types of models are sometimes called $\psi$-epistemic$^1$ and $\psi$-supplemented and are contrasted with $\psi$-complete models that interpret the wavefunction as a complete description of a physical state.
Wavefunction is a theoretical construct
Finally, someone making the assertion might be recognizing that physical theories are abstract systems of concepts and rules that aim to reproduce results of observations. Some of the concepts correspond to observations. Others, such as the wavefunction, make no direct contact with empirical reality but are included to make the theory work. In a sense, they represent intermediate steps in calculations. In this view, ascribing "actual existence" to them is moot. Consider for example two quantum engineers: Alice who is a "wavefunction skeptic" and Bob who is a "wavefunction believer". Faced with a task of predicting the output distribution of a quantum circuit they will both compute the wavefunction $|\psi\rangle$ as an intermediate step before applying the Born rule. They might tell themselves different stories about the quantity they computed. Alice might attach little meaning to $|\psi\rangle$, treating it as an intermediate step in her calculations. Bob on the other hand might regard $|\psi\rangle$ as something real even if unobservable. The key point is that regardless of the adopted narrative each computes the same outcome distribution.
Regardless of the ontological status we assign to the wavefunction, it is clear that to make the theory agree with observations the rules we use to compute it must account for the possibility of interference of different amplitudes. In other words, whatever the wavefunction is - something real, an incomplete picture of some other more fundamental reality or a reflection of our imperfect knowledge - it interferes.
$^1$ See PBR theorem and this paper for some issues with the epistemic interpretation of the wavefunction.