Reasons for a graduate student to attend conferences (in no particular order):
1. It presents you with an opportunity to present your research. I realize that the original poster is asking outside of this context, but you don't even have to present your research formally. Conferences can be a great place to sit down with other people over lunch and say "Hey, I'm working on this... I'm running into these problems... so far our preliminary results show... etc." There's a lot of value in just explaining your work to other knowledgeable people.
2. Networking. Conferences are where you can meet potential future employers. You can also meet other students or professors working in your area. Sometimes having someone you can email with a simple problem can save you weeks of work. Also, some conferences allow you to make industry contacts.
3. Learning. Conferences are where people present the current state of what they are working on - not what they finished six months ago that's just being published now. They are also an opportunity to generate context for many of the concepts that you may have learned about in courses. They can offer the opportunity for review and even the introduction of new concepts.
4. Fun. Remember, the whole reason that you got into your field in the first place was because you were passionate about it, right? I find conferences tend to re-kindle that passion for me on a regular basis. You get to spend a couple days focusing on your field, away from the distractions of your office. You also get the chance to travel. Conference organizers usually make some kind of effort to show off their city.
As to listing conferences on a CV - you may want to do that in the context of "continuing education" - probably not so much as a student, but as someone who had graduated. Some organizations, for example, will run an annual "summer school" or a "winter school." If you are in a profession like medical physics, for example, you can use those as one piece of evidence to demonstrate that you are keeping up with the field.