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.

Special relativity is necessary to give the mathematical formulation  .

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′.

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.

Special relativity is necessary to give the mathematical formulation.

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′.