I've had this explained to me in class, but I just don't get it. Is there a simple way to learn it? Thank you!
The best way to think about units is as if they were variables in an algebra equation. For example, if you had the variable "a" on the top and bottom of a fraction you would eliminate it because it cancels itself out. The same thing can be said for units. The picket fence outlined in your example is just a way of helping you set up your math problem so that you have units on top and bottom that ultimately cancel out. The "conversion factors" being used can be anything where what's on the top and bottom are equal. In other words, 1 km and 1000 m are the same thing, 2.54 cm and 1 in are the same thing, so on and so on. If then you wanted to know how many centimeters in a foot you would start with 1 ft then multiply it by (12in/1ft) because the "ft" on top and bottom would cancel (leaving you with inches). Then you would do the same thing with (2.54cm/1in), leaving you with centimeters. If you do the process in multiple steps it makes a little more sense. The picket fence method just allows you to string a bunch of conversation factors together in order to speed things up.
Hope this helps!