ARectangle rec = new ARectangle(1.0,3.0);
Won't print out anything. Your getArea() method returns a double to the main but since you are not displaying it or assigning it to anything so the main doesn't really do anything with it.
As for the toString() as mentioned by GregBrannon it is a way for the object to be displayed in the format you want it. If you don't include the toString() method and just use System.out.println(rec); it will give you something like ARectangle@be230d5.
In your initial post you mentioned doing "Area" + this.getArea(); What error did you get back then? Also keep in mind a toString() method must return a string.