How wave propagation is explained? Using phasors to represent [electromagnetic] waves allows to describe how some physical quantity changes with time in one specific point in space.
I'm interested to know how can be represented propagation of some physical quantity in space, and which underlying "mechanism" makes a quantity oscillating in one point in space be transferred to another point in space.
I'm interested in electromagnetic waves, but curious if the same explanation may be used for other kinds of waves.
 A: A wide variety of physical systems with wavelike behavior are described by the same partial differential equation, which is therefore famous as “the wave equation”. In three-dimensional space it looks like this:
$$\frac{\partial^2 f}{\partial t^2}=v^2\left( \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2} \right)$$
Here $f(x,y,z,t)$ is whatever quantity is “waving” — air pressure, electromagnetic field, gravitational field, etc. — and $v$ is a constant which turns out to be the speed at which the wave propagates.
What it says is that spatial changes in the waving quantity drive temporal changes. For example, if the air pressure in a room is the same everywhere, no sound wave propagates. But if there is more pressure in one spot than another, the excess pressure tries to even itself out. However, because of the second time derivative, this evening-out process overshoots the mark and causes an oscillation. This is because, at each point, when $f$ gets spatially evened-out, its acceleration becomes zero but its velocity is nonzero. So it develops spatial unevenness in the “other direction” of $f$... such as lower-than-average pressure rather than higher-than-average pressure.
Addendum in response to comment:
Since you're interested in phasors, note that
$$f=A e^{i(k_x x + k_y y + k_z z - \omega t)}$$
is a solution as long as
$$\omega^2 = v^2 (k_x^2+k_y^2+k_z^2).$$
It represents a plane wave moving in the direction specified by the "wavevector" $(k_x,k_y,k_z)$, with wavelength
$$\lambda=\frac{2\pi}{\sqrt{k_x^2+k_y^2+k_z^2}}$$ 
and frequency 
$$\nu=\frac{\omega}{2\pi}.$$
A: It is an elastic ineraction of neighbouring points that trasmitts the "signal" - a perturbation at one point is "felt" by the next point. The points must be "coupled with each other", otherwise, if they are independent, the perurbation stays at the same point.
The wave direction may not follow from the differential equation, but is determined with the boundary conditions: where and how you apply an external force and create the perturbation.
A: 
Considering electromagnetic wave, what are its "points"? Why the
  notion of electromagnetic "field" was introduced?

You are asking a very deep question. For simplicity, you may consider electric and magnetic fields as some sort of "elastic matter". They endeed carry energy-momentum, so it is not so far from the deep reality. The deep reality is such that these fields make sense as an external force in the equations of motion of charges ("receiver"). Depending on the receiver nature, the incoming wave may be absorbed (partially or completely) or "reflected". The receiver imposes the other boundary condition to the wave equation ;-)
