The diode doesn't prevent back EMF, it prevents the back EMF from doing damage.
"A coil" in this sense is referring to a device with inductance.
Inductance means that it has the property of resisting the change of current flowing through it. (If you understand inertia, it is somewhat analogous). Initially, when power is applied, this means that the current ramps up -- this is not a problem for your circuit. However, when the transistor turns off, the inductance wants to keep the current flowing.
Keeping the current flowing when the transistor is turned off doesn't sound possible. However, what happens is that the voltage across the inductor rises to allow the current to flow. This voltage will rise until sufficient current flows to expend the energy that has been stored in the inductor (incidentally, as a magnetic field).
In your circuit, the most likely thing is that the voltage will rise until the transistor breakdown voltage is reached, and energy will flow through the "off" transistor. This avalanche breakdown of the transistor is not a normal mode for most transistors, and can easily cause damage.
So how does the diode help? Simply speaking, it provides an easy path for the current to flow, minimizing the magnitude of the voltage spike that is created.
If you imagine the current flowing down from the +ve rail, through the buzzer, through the transistor, and finally to the ground (or -ve) rail. When the transistor turns off, the current continues to flow through the buzzer. Without the diode it has to find a path through the transistors and your power source. With a diode, it simply goes back up the diode to the other side of the buzzer.
If you are thinking of conventional current (which does from +ve to -ve), then the Arrowhead of the diode must point the other way (toward positive).