Here is how K-Cup machines work:
You insert the k-cup of your choice into the top of the machine, fill the water tank with water, select the size of the cup you want made and then start the brewing process.
The brewer then heats up the water and punctures small holes into the top and bottom of the k-cup. Hot presurized water is then forced into the cup.
Inside each k-cup is a paper filter holding the coffee and an air gap underneath between the filter and the bottom of the cup.
The water passes through the coffee and then through the filter to emerge into the gap at the bottom, where the holes punctured into the cup allow the coffee to emerge.
The beauty of this system is that the cups are pre-measured, so you can't accidently put in too much or too little coffee (using the wrong amounts is the main reason most home brewed coffee doesn't taste as good as that found in a coffee bar).
K-cups come in hundreds of different varieties of coffee (and tea and hot chocolate). Once your coffee is brewed, you simply remove the cup and dispose of it, which means there is less mess making coffee with this brewer than with other types of coffee maker.