The connection between this claim and the scenario on modern roads will depend a bit upon what country or region your roads are located in. There are very different customs of driving in different countries.
In the United States, a major connection is that we see on our roads the consequences of being "liberty drunk." Gardiner says that people who are liberty drunk do whatever they want without considering the needs of others and of the society as a whole. On the roads, they cut in and out of traffic or they eat or shave or apply makeup or send text messages as they drive. These actions are convenient to them, but they put others in danger.
Gardiner argues that society is weakened when people act in these ways. We can see this on our roads in the phenomenon of "road rage." As people act in selfish ways, our social cohesion breaks down. We then think that it is acceptable to act very aggressively towards other drivers (perhaps as Gardiner would like to act towards the man who was talking loudly and incessantly as he was trying to read). In this way, the scenario on our roads today shows us that people who are "liberty drunk" break down our social cohesion with their actions.
A.G. Gardiner defines the "rule of the road" in the following way: "It means that in order that the liberties of all may be preserved, the liberties of everybody must be curtailed." In other words, each person must have some limits on his or her freedom in order to enjoy the freedom that comes from social order. This order relies on everyone understanding how their actions affect those of other people. Gardiner claims that people are becoming "liberty drunk" and only recalling their liberties, not the responsibilities and limits that this liberty relies on.
Today, people often drive with a goal of arriving at their destinations as quickly as possible with, at best, little consideration for those around them and, at worst, a reckless disregard for others. There are frequent incidents of "road rage" and general disregard for rules (for example, people often go around people on exit ramps onto highways, not seeming to care that such an action is very dangerous). Many people's major goal seems to be to pursue their greatest degree of freedom in driving in the way they would like, but they seem to forget that their actions impose a lack of freedom on others. If their actions cause an accident, the order we all rely on to drive will be broken, and no one will get anywhere. People must submit to rules and to some limits so that we can all access the freedom that can only come from a condition of social order.