>And why from one frame of reference, you can see an electric field, and from another frame of reference you see a magnetic field - because it depends on your angle relative to the electron's motion?

No , it does not depend on the angle of motion, but on the [inertial frames.][1]

S[pecial relativity is necessary][2] to give the mathematical formulation .

[![E and B inertial frams][3]][3]

>Lorentz boost of an electric charge.

>Top: The charge is at rest in frame F, so this observer sees a static electric field. An observer in another frame F′ moves with velocity v relative to F, and sees the charge move with velocity −v with an altered electric field E due to length contraction and a magnetic field B due to the motion of the charge.

>Bottom: Similar setup, with the charge at rest in frame F′.


  [1]: https://en.wikipedia.org/wiki/Inertial_frame_of_reference
  [2]: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Transformation_of_the_fields_between_inertial_frames
  [3]: https://i.sstatic.net/JSOPG.png