A publicist books media interviews, and sometimes speaking engagements, for their clients. It takes tons of time and know-how to develop a list of media contacts, pitch to those contacts and follow-up with them.
There are thousands of potential media programs to contact, requiring research on their demographics and subject focus. Having a relationship with a contact is a big plus. Additionally, there is an art to pitching, knowing how to create a hook and tie it into the program's area of focus and current news events. And finally there is the very time-consuming task of following-up, which is a must since you are not likely to get booked for an interview after one pitch to a contact.
A publicist can be hired at any time, but is particularly helpful if you have a specific project to publicize, such as a book you have authored.