It means they're a popular person. :p
Usually they haven't cleared out their old messages in their voice mailbox.
It means exactly what it says... The phone only holds a certain number of voice mails, and when you get that, it meant they're at that limit and can't receive anymore until they delete some. It's not that hard to understand.
That they have so many voice mail messages, their inbox is at capacity and you won't be able to leave a voice mail until they clear them out.