How exactly does polarization by scattering work? Consider an electron sitting at the origin of a coordinate system. Let an unpolarized light travelling in the $z$-direction excite the electron at the origin. The motion of the electron can be thought of as two independent oscillatory motions, one along $x$-axis and the other along $y$-axis. 
If we look at the scattered radiation along $y$-axis, there will be none due to the motion along the $y$-axis. The scattered radiation that reaches the eye when viewed along $y$-axis is due to the motion along $x$-axis. It is true that an electron oscillating in the $x$-axis will give rise to maximum intensity when viewed along $y$-axis. 
I cannot understand why an electron oscillating along $x$-axis will produce electric field polarized along $x$-axis.
 A: Because the oscillating scattering electron behaves like an oscillating electric dipole in the sense that both can be represented as a small oscillating source of current.
The radiation fields due to such a system are described in any Electromagnetism textbook.
The oscillating charge acts like an oscillating current, backwards and forwards in the direction of oscillation. One then solves the inhomogeneous wave equation using its general solution, that tells us that the magnetic vector potential generated by the oscillating current is in the same direction as that current. The magnetic field is the curl of this vector potential and so is directed azimuthally curling around the oscillating current. The electric field of the transverse waves is then perpendicular to the magnetic field and also to the radial vector pointing away from the oscillating dipole - i.e. in a poloidal direction (the $\theta$ direction in spherical coordinates).
Thus whatever the viewing direction, the electric field lines up with the projected oscillation direction, with no component perpendicular to it. It is therefore linearly polarised and the polarization direction is $\hat{\theta}$.
Now in the case of the scattering example, if you have an oscillation along the x-axis and view that radiation along the y-axis, then the $\hat{\theta}$ direction is the same as the $\hat{x}$ direction.
This is a wordy explanation. The maths is more elegant, but can be found in most Electromagnetism texts.
