How does an electromagnetic wave move? Somewhere I found the explanation that the EM fields create and destroy each other during the oscillation (I suppose by Faraday's law) and this makes the wave "move". 
I can't imagine this because unitary vectors in E,B and k directions are a right hand ordered set of vectors and by the fact of E and B are in phase.
So, this is my question, what is the reason because the wave moves?
 A: Short answer
You've got half of the answer right when mentioning Faraday's Law of Induction, which tells us how electric fields can be generated from time-varying magnetic fields. The other half of the answer involves Ampère's Law with Maxwell's correction, which tells us how magnetic fields can be generated either from an electric current or time-varying electric fields. This coupled interaction between the electric and magnetic fields allows the EM waves to propagate.
Long answer
The previously stated claim is typically proven mathematically in most standard textbooks written on classical electromagnetism, and so such derivations can readily be found in the appropriate literature. I will give one such derivation here, adapted from the Wikipedia article that you can and should read in order to learn more about EM radiation in fuller detail:
Consider Maxwell's Equations in the microscopic form (in SI units),
$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \tag{1}$$
$$\nabla \cdot \mathbf{B} = 0 \tag{2}$$
$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{3}$$
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \tag{4}$$
with (3) and (4) being, respectively, the Maxwell-Faraday Equation and the Maxwell-Ampère equation.
In free space (i.e. in a location that contains no electrical charges or currents), the equations take the form,
$$\nabla \cdot \mathbf{E} = 0 \tag{5}$$
$$\nabla \cdot \mathbf{B} = 0 \tag{6}$$
$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{7}$$
$$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \tag{8}$$
Taking the curl of (7), we end up with,
$$\begin{align} \nabla \times \left(\nabla \times \mathbf{E}\right) & =  \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t}\right) \\
& = - \frac{\partial}{\partial t}  \left(\nabla \times \mathbf{B}\right) \\
& = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \end{align} \tag{9}$$
where we have made a substitution in the last step using (8).
From vector calculus, for a vector field $\mathbf{F}$, $\nabla \times \left(\nabla \times \mathbf{F}\right) = \nabla \left(\nabla \cdot \mathbf{F} \right) - \nabla^2 \mathbf{F}$, where $\nabla^2$ is the vector Laplacian operator, is an identity. Using this identity to rewrite (9),
$$\nabla \left(\nabla \cdot \mathbf{E} \right) - \nabla^2 \mathbf{E} = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \tag{10}$$
But since we're in free space, by equation (5) the first term on the left-hand side of (10) vanishes, and after redefining $\mu_0 \epsilon_0 = 1/c^2$ we're left with,
$$\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \tag{11}$$
which is clearly the wave equation with the electric field $\mathbf{E}$ as the dependent function.
Taking the curl of (8) and performing an almost identical procedure, the wave equation for the magnetic field $\mathbf{B}$ can also be derived,
$$\nabla^2 \mathbf{B} - \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0 \tag{12}$$
The fact that Maxwell's Equations in free space directly lead to these wave equations shows that the coupling between electric and magnetic fields described by Faraday and Ampère directly allows for the forward propagation of electromagnetic waves.
A: An electromagnetic wave is a change in the electromagnetic field of an object. The change in its electromagnetic field can only propagate at the speed of light. If the sun disappeared, it would take 499 seconds for the change in the gravitational field to propagate to earth and earth to go off at a trajectory; at the same time, earth would go dark due to the change in the electromagnetic field also reaching it at the same time. The further away from the source of the electromagnetic field, the weaker the field gets, so eventually the electromagnetic wave becomes too weak to detect. When an electromagnetic field is not changing, it does not induce a current in an antenna because the antenna is a capacitor, and current only flows when the voltage (charge of the capacitor) is changing, which is why voltage leads current by 90° – it is an AC circuit.
The change in the electromagnetic field strength clearly propagates as a wave outwards at the speed of light when you think about it. The field itself is made of photons.
