This is probably the part of jiu-jitsu which people concern themselves with way too much. People want to know when they're going to be promoted, why they haven't been promoted, what they need to do to be promoted, why other people have or haven't been promoted etc...
Most importantly here is that you shouldn't be concerned about any of these things. The only person who needs to think about these issues is your instructor, all you should concentrate on is training and improving. If you keep training you will eventually get promoted, whether it takes 6 months or 6 years doesn't matter.
But what does a belt represent? It's certainly no guarantee of anything, it's totally subjective and individual to your own instructor. So that's what it really means... it's a sign of your instructor's faith in your jiu-jitsu ability and most likely a judgement of your character. Each instructor will have their own reasons and requirements for promotion to each belt (training ability and attitude, competition record, personality etc); so just keep training and be happy whenever they do choose to promote you. Although one thing that will probably slow down your promotions, universal to all instructors, is asking about when you're going to get promoted.
I know that a lot of people get disappointed when they attend a seminar and don't get promoted, or when they see others get promoted who they feel they are equal to... maybe they have 4 stripes on their current belt, or they've just dominated at a competition or two, perhaps they've just been at their current belt level for a long time, or they have tapped a few people the belt above them in training. Whatever the reason, it's silly to be disappointed about not getting promoted because there are going to be many more times when you don't get promoted than when you do over the entire course of your jiu-jitsu training. It's not a criticism or negative judgement of your ability, it just means your instructor doesn't think it's the right time for you. Another step on from being disappointed is people who quit because they don't get promoted... now that is crazy. Someone doesn't get promoted so they quit, ensuring they will never get promoted!
So, what happens when you do get promoted? Why are some people so obsessed with it?
For the first question, pretty much nothing major! The only changes are... competition gets harder and under certain rulesets (IBJJF being the main one) you can do some different submissions. Really, that's it. Maybe some people in training will roll a bit harder with you, but really, in the grand scheme of things, nothing changes. You will still tap the people you tapped the day before, the people who tapped you the day before still will and the people who were tough, fairly even rolls still will be. So don't feel pressure to perform, if your instructor didn't believe you to be the level you've been promoted to, they wouldn't have promoted you (ignoring those instructors who promote for money!).
Now why do people obsessed with it? I guess it's a common human trait to like to be rewarded/congratulated, and there is nothing bad about this. I am sure some people enjoy a feeling of power or elevated status and this certainly is bad. Everyone should be equal in training, apart from the fact that the level of jiu-jitsu ability will vary. I do not believe any other sort of hierarchy is conducive to a good training environment... don't think that your belt means anything more than an indication of your jiu-jitsu ability.
Finally, if you do place a lot of importance on your belt colour, be prepared to be brought down to earth whenever you speak to people who don't know about jiu-jitsu. You will have conversations like these...
"So what belt are you now?"
"Is that a high grade?"
"No, one after white"
"So how long have you been training?"
"Oh"... cue awkwardness as they don't know how to ask you whether you suck.
"So are you still doing that jiu-jitsu thing?" (possibly accompanied by a karate chop motion)
"Yeah, I'm a black belt now"
"Oh cool, my 8 year old cousin has a black belt in Taekwondo"
My advice, just enjoy training and don't worry about your belt. I've never asked my instructor about my belt, chances of promotion, how long until the next belt or anything like that. I've just trained and always tried to improve, whenever I've been promoted I've been proud that my instructor feels confident enough for me to represent him at that level.