The ratsnest is the set of green lines that indicate unrouted nets (ie. nets that still need to be connected on your layout).
You can mouseover any ratsnest line to see what net it corresponds to.
The ratsnest will be hidden if the Design Rules layer is hidden.
You can hide individual ratsnest lines by turning them off from the Net List. This is particularly useful when you later plan to connect wide-reaching nets like power and ground through pours. In this case you may want to hide the GND ratsnest lines to reduce clutter.
While routing a trace, the corresponding ratsnest line thickens and follows the leading edge of the trace being routed. This helps you to constantly see the target destination you need to reach with that trace.
Ratsnest lines correspond to constraints violations that say "Net <net name> is not fully connected". When you fully route a net, both the ratsnest line and the violation disappear.
A ratsnest line will be drawn between two pieces of copper that are supposed to be connected according to your schematic. For example if you flood a pour that becomes split into 2 separate pieces, a ratsnest line will be drawn between them (and a constraints violation will occur) so that you know the connection is incomplete.
The ratsnest can be disabled entirely from the Constraints Manager.