People like to use the term "seeing someone" when their romance is budding.
This allows for them to let those around them know that they are extremely interested in someone else and not open to dating other people. Although those who are seeing each other are not automatically exclusive, couples who use this terminology to describe their relationship are headed in that direction. They are growing more and more fond of each other and are using each date to get to know each other a little better.
The term "seeing someone" also has a very strong physical component.
Couples who are seeing each other are typically involved physically, which is the cause for their desire to let others know that they are not interested in dating anyone else at the moment. This physical connection sometimes brings couples closer together and gives them the extra push that they need to go from seeing each other to beginning a full-fledged exclusive relationship.
People who are "seeing someone" are dating.
They go out on dates regularly with the person that they are seeing. They may attend lunches, dinners, movies or other activities that people in relationships typically enjoy. They often attend parties together and spend time alone at home together watching movies or hanging out. The whole point of seeing someone is that the person sees the other person often, usually at least twice a week, if not more.
It does not take long before couples who are seeing each other become exclusive, provided that as they continue to get to know each other better they realize that they are compatible.
Since the term "seeing someone" provides a sense of exclusivity already, it only takes one conversation between the couple to determine whether or not they are going to begin an exclusive relationship.