How Special Relativity causes magnetism
It is a mistake to think that special relativity (SR) causes magnetism. The article you're reading is arguing that
- (A) "If electricity exists, and SR is true, then magnetism has to exist".
This statement is true, I admit it. But it's equally true that
- (B) "If magnetism exists, and SR is true, then electricity has to exist".
In reality, electricity and magnetism are equally fundamental parts of physics. Special relativity unites electricity and magnetism into electromagnetism, in exactly the same way that it unites space and time into spacetime. Time does not cause space, space does not cause time, and SR causes neither space nor time. SR merely reveals the relatedness of space and time. Similarly, electricity does not cause magnetism, magnetism does not cause electricity, and SR causes neither electricity nor magnetism. SR merely reveals the relatedness of electricity and magnetism.
A lot of people come across (A) in their high school or intro college physics classes, and wind up misunderstanding it as the fundamental reason that magnetism exists. Why is that? And why do textbooks almost never point out (B)?
The reason is simply that nobody questions why electricity exists--electrical attraction and repulsion seem perfectly natural--whereas magnetism seems more mysterious. In other words, this pedagogical asymmetry between electricity and magnetism has nothing to do with physics, and everything to do with our inborn preconceptions and intuitions.
There are two phenomena in your question.
(1) Let us first understand how magnetic field can be considered to "arise" because of relativity. Imagine a frame of reference in which a charge $Q$ is at rest. If another charge $q$ is brought in its vicinity, it will experience only an electrostatic force. Now get on to another inertial frame of reference moving at a velocity $\vec{v}$ with respect to the first one. In this frame of reference, you will observe both the charges moving. The static charges of the old reference frame now appear as charges and currents. The electrostatic field of the previous frame now appears as an electrostatic field of different magnitude and a magnetic field. Since physics is the same in all inertial frames of reference, we are inclined to believe that $\vec{E}$ and $\vec{B}$ are manifestations of a single electromagnetic field.
This is a very "hand-waving" kind of an explanation. You may want to refer to Rober Resnick's "Special Theory of Relativity" or Melvin Schwartz's "Principles of Electrodynamics" for greater mathematical details.
(2) The first point tries to explain how magnetism due to a current can be considered to be a relativistic effect. Now let us consider magnetism due to electrons. Apart from charge and mass, electrons also have an intrinsic magnetic moment that can be explained only through relativistic quantum mechanics. Thus, magnetism of a bar magnet is also a relativistic effect. Please note that magnetism in a bar magnet is because of the electron's spin and not orbital motion.