Significance of crystal classes in crystallography
The references you and @farrenthrope cited (or a good introductory mineralogy text) should have answers. 

In brief:

The crystallographic class is significant to mineralogy because it is direct result of the mineral's crystalline lattice (its most fundamental property.) Each mineral can belong to only one crystallographic class.   

How can I determine the [crystallographic] class an [unidentified] mineral [specimen] falls under?

You'll need to inspect your specimen and try to recognize the symmetry operations (like rotations and reflections) present in some physical properties of your specimen, commonly the symmetry of euhedral crystal habit, the symmetry of optical properties, or the symmetry of an x-ray diffraction pattern etc. Each crystallographic class is defined by a unique set of symmetry operations. 

How were the 32 classes determined?

A combination of observation and mathematics. The law of rational indices was deduced by Haüy (1784, 1801) from the observation of the stacking laws required to build the natural faces of crystals by piling up elementary blocks, for instance cubes to construct the faces of the rhomb-dodecahedron observed in garnets or the pentagon-dodecahedron observed in pyrite, or rhombohedrons to construct a scalenohedron of calcite. Hessel (1830) proved that, as a consequence of the law of rational intercepts, morphological forms can combine to give exactly 32 kinds of crystal symmetry in Euclidean space, since only two-, three-, four-, and six-fold rotation axes can occur.