Why does a moving electric charge creates a magnetic field around it?
The Biot-Savart law describes the conversion of the electric field of a moving charge to a magnetic field. It shows that a moving charge will create a magnetic field rotating around it based on the "right-hand rule" with the strength of the magnetic field increasing proportionally to the charge and velocity of the particle and decreasing proportionally to the square of the distance away from the charge.
Maxwells equations also account for the motion of a charge creating a magnetic field, albeit in a much more complicated way.
However, if your question is "Why" does it create a magnetic field. The answer is pretty much "Nobody knows!" We have just observed this effect experimentally by methods like the iron shavings mentioned above or by measuring the current generated in another wire as a result of the magnetic field.
I know that can't be satisfying, but it's the reality of the situation. It's difficult to explain why exactly something is the way it is when we're trying to explain a basic point of physics. This happens in a lot of fields in physics, though. Why is the speed of light 3 * 10^8 m/s? Why did relativity have to be correct instead of classical physics? We certainly don't know now, nor are we likely to anytime soon!
When a charged particle—such as an electron, proton or ion—is in motion, magnetic lines of force rotate around the particle. Since electrical current moving through a wire consists of electrons in motion, there is a magnetic field around the wire.
This field can be demonstrated by placing fine iron filings or shavings on a car and sticking a wire through the middle of the card. When a DC electrical current is passed through the wire, the iron filings align to show the magnetic field.
Iron filings show magnetic field around electric wire