Simply put, a gadfly is a horsefly. Anyone familiar with this insect would be aware of its irritating buzz and sharp bite. In our antiseptic age of screened windows, exterminators, and self-contained environments most of us are relatively free of the gadfly's persistence.
In the ancient world, however, this pesky insect would be well known to
Socrates' audience (keep in mind how cities,
especially those without modern sewers and refrigeration for food, attract
flies). Like the horsefly, Socrates "buzzes"
and "bites" at the complacent, compelling them to consider matters of virtue.
Also like the horsefly, Socrates is likely to
be "swatted away" (i.e. killed by the state)!