Scoring caulk means that you sever the adhesive connection between the caulk and the surface to which it is applied. When you caulk a bathtub, for example, the caulk will adhere to the bathtub and surrounding walls when you install it.
To make your caulk removal job you will need to score the caulk first. Wet the caulk to make it easier to cut, and then use a utility knife to cut along the edges of the caulk to separate it from the surface.
You could spend more money on a specialized caulk scoring tool, but an inexpensive utility knife will do the same job.
Most people have a putty knife sitting around the house, and if you do not have one then it is very inexpensive to go out and purchase one. Once you have scored the wet caulk you can use a putty knife to pry it up from the surface and pull it away.
Much of the caulk will come off in large pieces, but for the more persistent adhesive you only need to wet it again and then use the putty knife to finish the job. An inexpensive putty knife will do the same job for you that an expensive specialized caulk removal tool would do, and if you have a putty knife with a sharp enough edge, then you can use your putty knife to score the caulk as well.
That would allow the inexpensive putty knife to do the job of two specialty tools.
Part of removing caulk is cleaning the surface and making it ready for a new line of sealant. Over the years, caulk can cause a small build-up of mold and other materials that should be cleaned away completely.
Add a cup of pure ammonia to a half gallon of warm water, and then use a sponge to clean the area where your caulk was. In most cases you will have ammonia in your home to use with your laundry.
You can also use that ammonia as a cleaning agent to help finish your caulk removal project.