The traditional Otto cycle used four "strokes", of which two can be considered "high power" – the compression and power strokes. Much of the power lost in an engine is due to the energy needed to compress the charge during the compression stroke, so systems to reduce this can lead to greater efficiency.
In the Miller cycle the intake valve is left open longer than it normally would be. This is the "fifth" cycle that the Miller cycle introduces. As the piston moves back up in what is normally the compression stroke, the charge is being pushed back out the normally closed valve. Typically this would lead to losing some of the needed charge, but in the Miller cycle the piston in fact is over-fed with charge from a supercharger, so blowing a bit back out is entirely planned. The supercharger typically will need to be of the positive displacement kind (due its ability to produce boost at relatively low RPM) otherwise low-rpm torque will suffer.
The key is that the valve only closes, and compression stroke actually starts, only when the piston has pushed out this "extra" charge, say 20 to 30% of the overall motion of the piston. In other words the compression stroke is only 70 to 80% as long as the physical motion of the piston. The piston gets all the compression for 70% of the work.
The Miller cycle "works" as long as the supercharger can compress the charge for less energy than the piston. In general this is not the case, at higher compressions the piston is much better at it. The key, however, is that at low compressions the supercharger is better than the piston. Thus the Miller cycle uses the supercharger for the portion of the compression where it is best, and the piston for the portion where it is best. All in all this leads to a reduction in the power needed to run the engine by 10 to 15%. To this end successful production versions of this cycle have typically used variable valve timing to "switch on&off" the miller cycle when efficiency does not meet expectation.
In a typical Spark Ignition Engine however the miller cycle yields another benefit. Compression of air by the supercharger and cooled by an intercooler will yield a lower intake charge temperature than that obtained by a higher compression. This allows ignition timing to be altered to beyond what is normally allowed before the onset of detonation, thus increasing the overall efficiency still further.