We say that light has magnetic and electric fields ossilating perpendicular to each other.What causes the those fields to exist and why does the light as a whole does not get affected in the presence of other charges or magnetic fields.
The electromagnetic fields are generated by the motion of an oscillating charge (for example, electrons in an antenna for RF - radio frequency - waves; or electrons bound to an atom for light). The relationship between the oscillating charge and the fields are given by Maxwell's equations.
Because these fields are oscillating, there is no net influence on them if you put them in the presence of another (DC) field. In principle it is possible for photons to interact at sufficiently high energies (see two-photon interaction) - this proves that it is just about possible for an electromagnetic wave (photon) to interact with another. But that's well beyond the level of your question.