How does adding more coils to an AM tuner coil allow it to tune Shortwave?
The diagrams show the new coil being added electrically in parallel to the existing coil.  This makes the total inductance lower.  

The new coil has a very low inductance (very few loops.)  The existing coil has many loops.  Put in parallel, the total inductance is lower than that of the low inductance loop.

Lower inductance means it passes higher frequencies better.  Short wave radio is transmitted on much higher frequencies than regular AM radio.

Shortwave starts a little over 2MHz and goes up to around 30MHz.

Regular AM radio is between 530kHz and 1.6MHz.

So, when you put in the new, low inductance coil, you make the radio more receptive to higher frequencies.