The objective of a liquid-liquid extraction is first and foremost to separate two or more components of a solution from one another. Doing so not only allows you to perform further analyses and procedures on either or both of said components, but also allows you to determine whether or not a certain compound is present in an unknown reagent. For example, if you have a light blue, opaque solution and want to know whether or not it contains nonwater-based components --- "alkanes," in chemistry jargon --- add a water-based compound to see if any separation occurs.
When two or more layers of liquid separate within a reaction chamber, you can instantly observe many of their physical properties. Perhaps most obvious --- and, depending on your experiment, most important --- among these is density, a metric which reflects how densely molecules pack themselves within solution. The component which floats on top of the other, by definition, has a lower density. Once separation occurs, you can also observe properties and characteristics such as color, opacity and the proportion in one which one solution exists in comparison to the other.
Another potential objective of liquid-liquid extraction is the isolation of one or more reagents for later use. This can be particularly useful for solutions you don't have on hand in their pure form. After separating layers, you can use a medicine dropper to draw off the top solution, either for use or to expose the bottom one. At this point, you can use the reagent of your choice as a reactant in a subsequent experiment or run further analyses on it.