Let us start from the beginning, what does physical mean and what does "mean" mean:
The physical meaning of electromagnetic waves
The language of physics we learn not only at school but by everyday experience, is a language developed to describe the behavior of nature for dimensions larger than 10^-9 cm , and times larger than 10^-9 seconds; in the present day physics language, dimensions large enough that the Heisenberg Uncertainty Principle (HUP) is automatically fulfilled. There is also a mathematical description for students of physics.
So we have given a " mathematial meaning" to a sea wave , by modelling its motion in transfer of energy with sinusoidal functions which are solutions of differential equations.
Then we have instruments which discover physically that at the small dimensions of the HUP there exists a substructure of atoms and molecules. The enlightenment coming from this experimental observation is what powers the "physical" in the title question. It considers "physical" something that can have an experimental under layer which layer can be modeled mathematically again.
As a highschool teacher I had used to say: "problem number one has been reduced to problem number two". I.e., "what is the physical meaning of atoms and molecules" . Again an underlying level was revealed by experimental observations, the quantum mechanical models of atoms and molecules, and again the problem of what is "physical" has been pushed a layer below.
From the above, we can see that the term "physical meaning" is defined by the existence of an experimentally measurable under-layer that explains the behavior under study, pushing the "unknown" a layer below. We are making great efforts with our colliders etc to find lower and lower layers, if they exist. At present we are stopped with the standard model of physics which has pushed the under-layer to the posited existence of the particles in the table, the elementary particles, which mathematically are modeled as point and are the building blocks up to all layers I have been discussing above.
Notice the photon in this table. It is the building block of the electromagnetic wave, so it is the under-layer of the electromagnetic wave. That is the physical meaning of the electromagnetic wave, that it emerges, its mathematical models, from a confluence of photons. This needs quantum field theory to be understood, but it is just mathematics, as our experiments have not reached an under-layer which will be modeled mathematically so that the photon can be considered a measured confluence of the under-layer, the way the molecules are an under-layer for the water wave.
So, what is an electromagnetic wave?
An over-layer emerging from the confluence of photons.
How can it move vibrating across the vacuum? How can we physically interpret phenomenon of electromagnetic waves?
It is riding on an enormous number of photons . The mathematical description of this can be seen here.
We are stuck with the mathematical meaning because photons themselves are the bottom layer that can be measured in our experiments. That means we are stuck with quantum mechanical mathematical models describing the photons, which correctly gives the E and B fields of the classical wave and their energy transport in space. We have not reached experimentally the point of seeing whether there exists an under-layer to this or not, i.e. whether we have reached the end of the line with the standard model and its extensions as far as describing nature and giving meaning to "physical".