We bet you're wondering about all the references to horses, reigns, and bits. You're probably also wondering what the heck horse-related metaphors and imagery are doing in a play about whether or not sexuality can be policed by the government. Let's discuss.
When Duke Vincentio talks about his tendency to be a lax ruler, he speaks about his subjects as though they're a bunch of "headstrong" horses that need to be reined in by Vienna's laws:
We have strict statutes and most biting laws.
The needful bits and curbs to headstrong weeds,
Which for this nineteen years we have let slip; (1.3.3)
The Duke admits that, for the past several years, he's allowed his unruly subjects to flout the laws of Vienna. Specifically, the Duke's allowed his subjects to thumb their noses at the city's sex laws, which is why prostitution and fornication have become uncontrollable.
As it turns out, this passage sounds a lot like what the sixteenth-century English Puritan Phillip Stubbes famously writes in The Anatomy of Abuses (1587). Check out the metaphor Stubbes uses when he complains that parents who don't punish their children are responsible for all of society's problems:
Give a wild horse the liberty of the head never so little and he will run headlong to thine and his own destruction also. [...] So correct Children in their tender years.
In Measure for Measure, this unruly human being = wild horse concept turns up repeatedly. When Angelo propositions Isabella, he says "I have begun / And now I give my sensual race the rein" (2.4.24). In other words, Angelo sees his pursuit of Isabella as a "sensual race" and says he can't control himself, which is why he gives in to unbridled desire.
This seems like a pretty fitting metaphor for the way human desires can either be put in check or given free reign, don't you think? For centuries, sex, lust, and passion have been associated with wild horses, which is why we so often see scantily clad men and women riding bareback in steamy music videos and perfume ads.
P.S. There's a similar metaphor at work in The Taming of the Shrew, where Petruchio's "taming" of "wild Kate" is often compared to a horse being broken in.