Polarization of a transverse wave travelling in ionosphere with polarization direction perpendicular to earths magnetic field Assume a transverse electromagnetic wave entering ionosphere such that its electric field is perpendicular to Earth's magnetic field. Now, I read that as it will enter plasma, the wave will tend to be elliptically polarized.
In other words: if Earth's magnetic field $B$ is in $z$-direction, electric field $E$ of the wave is in $y$-direction and propagation vector $k$ lies in $x$-direction, then it says that $E$ will develop a component along $x$ too.
How does that happen?
 A: One thing to consider is whether the wave, by being transverse, is linearly or elliptically polarized.  If it starts out as a linearly polarized wave and converts to an elliptically polarized wave (which can happen), that is different than circular to elliptical.
One also needs to consider whether the wave starts in the neutral atmosphere of Earth or in the ionized magnetosphere.  Think of Snell's law and the different response one finds when going from low(high) to high(low) indices of refraction.  The distortion of the wave vector, relative transmission vs. reflection, and attenuation rates are all different.
The type of mode is also important.  If, for instance, one considers a whistler mode wave, then those modes have specific properties that cause them to strongly refract when encountering density or magnetic field gradients in a plasma.
In your case it sounds like a combination of refraction and interaction (with the charged particles) occurs.  If $\hat{k}$ $\parallel$ $\hat{x}$ initially, then the phase fronts are moving in the $\hat{x}$ direction.  This is the direction that defines how the waves interact with the particles.  If the wave starts to couple to the plasma (e.g., it causes electrons to slosh around rather than passing through without interacting), then the motion of the particles can cause local time-varying charge separations, which are electric fields.
So from the little information given, my best guess is that the wave is starting to couple to the particles and that interaction causes particles to oscillate which produces an additional component to the electric field.  The wave probably refracts across the boundary as well, but the interaction with the particles, I think, is the key to your question.  I am also thinking, now, that you implied the wave started in Earth's atmosphere and propagated into the ionosphere rather than the converse.
