You see a rat's nest when you are working with PCBnew to create the artwork of the PCB.
This page has been checked to see that confusion in my brain (before 9 November, 2011) about the roles of pin names and pin and pad numbers hasn't led to errors here. I have a page with more on this if you are really curious.
The "rat's nest" (to a PCB designer!) is the collection of straight lines between pads. These lines are dynamic… as you move a footprint, the lines keep changing so that they still connect pads, and still remain straight. This behavior is aptly named "rubber banding"… it is as if each line was a rubber band, showing a pair of pads which need connecting.
That's the good news! The bad news is that in even a modest circuit, there are a lot of lines! And unlike the tracks which they will help you place, they do not eschew crossing and re-crossing other lines of the rat's nest.
Don't be fooled by rat's nest lines. The one from, say, pin 1 of your LED does not show you, directly, every pad that pad should connect to. It may be that you can create a PCB which implements your circuit by taking pad 1 of your LED someplace else, which itself eventually connects to the pad the rat's nest line goes to.
This is not a problem. As you lay down tracks, some of the rat's nest lines will go away. Furthermore, as you work, at any given moment, all of the pads which, eventually, must be connected will be highlighted. you won't have trouble finding "the best" track routing, and, even though they don't show you all possible solutions, the rat's nest lines are enormously helpful. In particular, in conjunction with the design rules checking tool, they help save you from overlooking necessary connections.