This may not be easy, but we'll give it a go.
I'm assuming that you mean with respect to an electromagnetic (or "EM") wave, and that by "phase difference" you mean the polarization of the electric and magnetic field components; if this isn't the case, then I'm not sure what you're talking about, because the the *phase* of these components (in terms of their amplitude vs. time) is NOT 90 deg. apart (instead, they are perfectly in phase).
Now, to figure out all of this through a rigorous mathematical treatment takes some doing, and is typically the subject of the better part of an introductory course in electromagnetics. So we're not going to do that here. But to visualize the situation, consider the plain ol' dipole antenna
- overall, a half-wavelength long, with the feedpoint in the middle (which splits the antenna into two quarter-wave elements which lie along the same line).
Such an antenna can be viewed as an open-ended quarter-wave section of transmission line, "opened up" (i.e., the two conductors of the line separated, forming the two quarter-wave elements). At the feedpoint (one-quarter wave back from the "open" end - or an "infinite impedance load" - in the original transmission line section), we have a very low-impedance point; this is, on that same original line, a point of current maximum and voltage minimum. The open end of the line - which, when "opened up" into the dipole antenna - is a voltage maximum and current minimum (it better be, since the load impedance there is really high, remember!).
Now visualize what this means in terms of the "standing wave" pattern on the antenna. You have a current maxmimum point right in the middle (at the feedpoint), and the current is moving along the elements, right? You also have a voltage maximum at the end points - the tips of the elements - and since the two ends are opposing one another, what this means is a large potential (voltage) ACROSS the antenna, end-to-end. Got it?
So think about the fields that result from this. The electric field generated by the big potential across the antenna is aligned with the elements of the antenna - i.e., the "field lines" would be visualized as connecting the tips of the elements. On the other hand, you know that magnetic fields which result from current in a conductor are visualized SURROUNDING that conductor - so view the magnetic field as encircling the antenna elements.
And voila, there you are - crossed electric and magnetic fields! Which, in this case, are being produced in such a way that the result is an EM wave being launched from the antenna, which was the whole idea in the first place.
Bob M.