How is the brain capable of seeing a spectrum of color when the eye only contains three types of color detectors?

By integrating them in the optical cortex. The three types
of receptors peak at yellow, green and red and are thought to process the full field of
color through opponent-process theory. For this theory to work, each color will activate
and inhibit a given receptor, and having it activate the correct receptor will give an
inhibitory response to the other receptors.  Red excites the violet receptor but
inhibits the green receptor, for example.  This is how the eye sees red.  Blue and
yellow are opposites too.  The brain integrates all the stimulatory and inhibitory
impulses and mixes them together in order to determine what color you are looking at.
 Different strengths of receptor activation result in the perception of different colors
as the brain combines and integrates the incoming signals from the retina.  Colorblind
people have a problem with the processing in that the cells in the brain that tell the
difference between red and green are missing.  There are cells in your brain called the
parvocellular cells that process the differences in activation between the green/red
receptors, and another type that processes the blue/yellow
difference.