In the days of horse-drawn carriages, there there were all kinds of words and phrases for different kinds of carriages, different systems of harnessing more than one horse. I usually just let the terms glide by and say to myself, "some kind of horse harness" and determine, from context, what the social implications were.
In this case, the word "smart" means clean, healthy, and well-groomed and "high-stepping trotters" means the horses were trained to trot in some fashionable way, so this is a wealthy person's carriage, the horse-drawn equivalent of a luxury car.
If I get curious, I need to look up the terms somewhere. The term "four-in-hand" is much more common, and actually is in the first dictionary I consulted:
"1. A team of four horses controlled by one driver.
2. A vehicle drawn by four horses.
3. A necktie tied in a slipknot with long ends left hanging one in front of the other."
I think it's a pretty safe guess that a two-in-hand is a carriage drawn by two horses, with the reins rigged in some specific way. A Google Images search of "harness two-in-hand" turns up a few pictures.