The most common reason, particular for lower-level jobs, is simply to save time and effort. Many organisations don’t have the staff to sift through all the applications that can be generated by an ad in a newspaper or on the Internet, so they use an agency to bring the numbers down to a manageable level or even fill the post for them.
For very senior or highly specialised appointments, employers will also often use a recruitment agency that specialises in finding people for that specific profession or level of experience, one which has good contacts in that field.
The third reason is confidentiality. The employer may be a high-profile organisation which wants to avoid the publicity that a particular job advert can attract or an avalanche of unsuitable applicants attracted by their name.