In simple terms, clipping path is an outline of an image produced by using the Pen tool of Photoshop. It is a closed vector path and it can be used to cut out areas of 2D images and also remove the background from the selected areas. The benefit of using clipping path is that it the path is reusable and transferable across images. If an image were to be cutout without a clipping path then if any mistakes were made, then it could not be undone without having to cutout the image from the start again. Using clipping path reduces the time to fix errors as the path can be easily modified and the area can be cutout from the original image again with just a few clicks.
Why would one require clipping paths? Well, there are a lot of reasons. Companies might want to display their products on a white background on their websites but their photos were shot on a background of a different color. Maybe you want to change the color of a certain section of an image so you would need to make a clipping path to select that area. There are many, many applications for this.
There are also entire companies dedicated to providing the clipping path service and also background removal service. Teams of a few individuals to a few hundreds are not uncommon. So it is very well possible to make a living from doing clipping paths as well.