The most familar physical example of SSB comes from ordinary ferromagnets (the things stuck to your fridge). For fairly subtle quantum mechanical reasons atoms in a ferromagnet tend to line up with their magnetic fields all pointing in the same direction. Most atoms don't do this, which is why most materials in the universe (ducks, toast, your little sister) don't stick to your fridge. In other words, if one atom is surrounded by other atoms all with fields pointing in the opposite direction, it will be energetically favorable for it to flip its direction and align with its neighbors. Regions within a magnet where the atoms are aligned and a net magnetic field points in a certain direction are called magnetic domains. An iron bar then becomes magnetized when all of its domains point in the same direction. Alternately, you can de-magnetize a bar magnet by heating it or hitting it with a hammer, which has the effect of randomizing the magnetic domains.
SSB occurs when an initially randomized state (physicists use the term 'symmetric' for this, even though it's a little confusing) spontaneously evolves to a special state. That is, the system moves from a state where the domains all point in random directions to one where they are all aligned. Or to push my analogy further, from a situation where countries randomly choose which side of the road to drive on, to a situation where everyone drives on the left. The point is that it doesn't fundamentally matter which side of the road we drive on, or which way the magnetic field points, but that often there are benefits to going with the flow. And whenever there are such benefits, nature devises ways to take advantage of them. This phenomena is found everywhere in the natural world, from the electro-weak transition to super-conductors to deciding which water glass to drink out of on a circular table.
It turns out that the reality of which nations drive on which side of the road is a little more complex than our initial short list of first-world nations would indicate. As can be seem from the map, many of the world's island nations are indeed left-driving countries, but there are also significant pockets of left-driving mainland countries that do share borders with right-driving countries. Most of these countries (India, Pakistan, Bangladesh, east Africa), share a common administrative heritage: the colonial British Empire.
The history of left vs. right driving nations is long and fairly interesting (both links give a good overview). There has been a significant trend over the 20th century towards alignment in these regards, even among former British colonies. A century ago, many more countries drove on the left, but with the increase in car traffic over the years, pressure has mounted to conform to right-side driving. Many nations (including Canada, China and the parts of Europe not invaded by Napoleon) have switched from left to right in the past 100 years, but the only recorded switch from right to left is the island of Okinawa in 1978. Leaving island nations aside, there are really only two domains opposed to the dominant right-driving domain: southeast Africa and the Indian subcontinent (I bet the border crossings there are always exciting). It may be that these regions are populous and (relatively) powerful enough to resist assimilation by their neighbors, or it may be that as car-traffic over the borders increases with time, they too will join the right-driving universe.