Summary. Positive-deviation azeotropes mean that like interactions are much stronger than unlike interactions, whereas negative-deviation azeotropes mean the opposite. The former case is more common because most substances interact better with themselves than with other substances---like dissolves like.
(So I've only just rephrased the question at a lower level: one could equally well ask why like should dissolve like, but believing this statement on faith is perhaps more palatable than just accepting that there are more positive-deviation azeotropes.)
First, a few definitions.
A binary solution showing positive deviations (negative deviations) to Raoult's law is one in which the dew-point curve in a pressure vs. mole fraction graph lies above (below) the line connecting the pure vapor pressure of the two species.
A binary azeotrope is positive deviation (negative deviation) or minimum boiling (maximum boiling) if it shows positive deviations (negative deviations) to Raoult's law and exhibits a local maximum (minimum) in the dew-point curve in a pressure vs. mole fraction graph. From solution thermodynamics, one can prove that the vapor and liquid mole fraction of either species is equal at these extrema.
Physically speaking, positive deviations (negative deviations) from Raoult's law imply that like interactions are stronger than (weaker than) unlike interactions in the solution, for the solution's dew point pressure is greater than (less than) that of an ideal solution.
Azeotropes are extreme forms of these positive deviations (negative deviations), when the dew-point curve further exhibits a maximum (minimum), such that at some composition the solution's dew point pressure is actually greater (lesser) than the pure vapor pressure of either species.
We expect, therefore, that positive-deviation azeotropes are more common than negative-deviation ones because it is more likely for two species to have stronger like than unlike interactions.