Why electric field and magnetic field are in phase in electromagnetic waves? I can understand in mathematics but could you please explain it's physical significance (such as from the energy perspective)?
 A: First of all I should say that I am a bit wary of separating arguments as "mathematical" and "physical". Physics ${\it is}$ all about applying math to the actual world. Often when asking for a "physical" argument, people is fact ask for an analogy with something they already know.
That being being said, let me add my 50 cents to the discussion above. First of all, having energy density becoming zero at some points in space does not mean the energy is not conserved. In one comment you are asking "...where the energy goes when E and B comes zero?". The answer is that it flows away. Remember that energy is only conserved ${\it globally}$. Locally the energy density $\rho_E$ is just subject to the continuity equation
$$\frac{\partial \rho_E}{\partial t}+div \left(j_E \right)=0$$
So no paradox here.
Also, as pointed out several times above, there is nothing special or important in the fact that energy density is modulated in space. The most trivial example is the circularly polarized EM wave. The the magnitude of both E and B are always constant and the energy density is uniform throughout the space. You can think of the spatial and temporal distribution of field and energy in a linearly polarized EM wave as just a funny effect of interference between right- and left-hand polarized waves.
A: Not always true. For example, inside metals electric and magnetic field of the EM wave are out of phase: magnetic field lags behind.
In other cases is quite easy to understand. Changing magnetic field $(\frac{\partial B}{\partial t})$ creates a changing electric field and vice versa (Look at the two Maxwell equations involving curl of E and curl of B) .
Let's consider the electric field part of light: It has the form: $E=E_0 cos(kx-\omega t)$, we need to look at the change as time passes so let's just look at some particular value of x and see how $E$ changes at that point. For simplicity let's say the point we choose is $x=0$. Now the field at $x=0$ is $E_{x=0}=E_0 cos\omega t$. It's derivative w.r.t time will be zero when cosine is maximum(+1) or minimum(-1), at all these points in time $E$ is constant. $\therefore$ the corresponding $B$ should also be a constant in time at these same instants. i.e, the peaks and valleys of $E$ and $B$ should coincide in time. This should be true not only for $x=0$, but for every $x$.
A: The statement often used for electromagnetic waves "changing magnetic fields create changing electric fields" is to say the least misleading as far as electromagnetic radiation goes.


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. The electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together.

Images of the mathematics as above go against our conservation of energy instincts.
It is important to remember that the Maxwell theory which results in the above model for electromagnetic radiation, defines energy with the Poynting vector.

The rate of energy transport per unit area is described by the vector


$E_m$ is  the maximum of the plane wave,  in the above  case.
So in the mathematics of Maxwell's solution the energy is carried by the averages of the given mathematical form.
I keep insisting on "mathematics" because the figures as shown cannot be measured, they are fits to measured data in optics but itself.
The behavior of energy of a light beam becomes clear when the photon content of the beam is considered, where each photon carries energy $hν$ and our intuition of conservation of energy can be fulfilled.
See this classroom experiment to get a feeling of how photons build up the classical electromagnetic wave.
A: . . . . . but could you please explain it's physical significance (such as from the energy perspective)?
In the mathematical derivation to show that the electric field and the magnetic field for plane waves are in phase a number of assumptions are made.
One of them is that the medium is non-dispersive with no loss of energy due to absorption ie the refractive index of the medium is real.
In a dispersive absorbing medium, the refractive index n is complex and so when the wave passes through such a medium the phase between the electric and magnetic field is no longer zero.
The loss of energy from the wave as it passes through a medium is coupled with a phase shift between the electric and magnetic fields.
Other examples of the fields not being in phase are: standing waves where the phase difference is $\pi/2$, evanescent waves where the wave vector is no longer real even in a non-dispersive medium, the near field around an antenna, . . . . .
