What's it all about?
When you work in the states, you typically get 2 or 3 weeks of vacation per year, and those vacations are usually whirlwind trips "somewhere else." You only see the broad stroke of the tourist industry in whichever country you visit.
But Slow Travel is just the opposite - slowing down, taking your time to experience a place. The general idea is to live as much as possible like the locals of the place you are visiting. Naturally, a longer visit allows more - you can rent an apartment, shop at the farmer's market, and go to county fairs. If the language is different, you can learn some. You can take a class in art, talk to your new neighbors about their culture, and learn something about local cuisine.
Planning for slow travel is almost as fun! Finding out about the politics, history, folk tales, songs, and art tells you a lot about your new "temporary country." Even learning some of the phrases necessary for politeness gives you a head start!
Slow travel doesn't even have to be in another country. Just consider how very different the culture is in various states in the USA - while the language is still English, people use it so differently. In much of the South, for example, children refer to adults as "Sir" or "Ma'am." In the much of the Midwest, everyone at a table waits to start eating until everyone else has food in front of them. The West is more casual in both eating habits and address - eat when you get your plate, and call people by first names. In Boston, "casual Friday" may mean men wear dockers and a polo shirt. In Los Angeles, you wear shorts if you go into the office on a weekend.
Those are the sorts of things you know just from talking to people, watching television, and reading novels. Think about how much more there is just waiting to be noticed!