The first scientists to explore the use of antibodies for medicine used mice, and sometimes other animals, to produce antibodies, particularly because that was one of the first methods available using the technology and information at the time. Now, there are different ways of producing antibodies, and in some cases no longer need to use mice.
Mice were used as the process involves removing spleen cells (immune cells), which are the cells that produce the specific antibodies you are looking for as a reaction to providing the antigen you want to target. As mice are used a lot in research, there are also many mouse breeds and cells available, such as the 'tumour cells' that are fused with the immune cells to produce new rapidly dividing antibody-producing cells, which are sometimes put back in the mice to grow, and then must be extracted. One disadvantage, however, is as the cells are from mice, parts of the antibodies will look particularly foreign when given to humans, and an allergic reaction may occur, resulting in side effects.