Over the course of a game your character will get into fights, conversations, relationships, and a whole ton of other things. They'll go shopping, they'll go adventuring, hell, they'll probably take at least one trip to a temple or graveyard over the course of the campaign. How they can interact with those things are determined by the rules of the system you are using. However, what determines how well a character can interact with those things is determined by the character's personal mechanics, which are jotted down on their character sheet.
The sheet is literally a treasure trove of information on the character. Not counting narrative details such as name, height, gender, weight, etc you also have the game's mechanics: stats, skill ranks, acquired gear, money, feats, advantages, powers, and so on until you've covered everything that the game has to offer. The system tells you to roll a number of dice equal to your stat + your skill. The character sheet tells you how many dice that actually is. See the difference?
Your character sheet, however, will also tell you other things. An advantage may give you a bonus to certain rolls, or a penalty. A certain level of skill may let you reroll 9s. An item in your inventory may mean you don't even have to roll in this particular instance. That knowledge is only recorded in one place, which is your character sheet, and you are - unless the GM has a back up copy - the only one in possession of it during the game.
So, why is it so important to know your sheet? Well, for one, it speeds up the game. A player who knows their sheet knows what bonuses and penalties they have, how the character works, and can have things ready to go with a little prompt from the GM. As opposed to a player who needs someone to go through and remind them of all the things they have each time, which slows the game down. For two, it increases your chance of survival. Like I said above, only you have the character sheet right in front of you, and it is not the GM's responsibility to remember that you have a +5 to all listen checks.
The most important reason though? In my opinion it is the relations at the table. See, if you forget your penalties, you are technically cheating. Not necessarily on purpose, but the rules say you have a penalty and odds are you've been paid in some way for it, (extra points, another ability, something) but you are ignoring the penalty. Depending on the group, this isn't a super horrible thing but it is something to keep in mind. I've seen people get resentful over the repeated 'forgetting' of penalties, and after the third or fourth time it gets pointed out people start to think - especially if it is negatively affecting them - that it is being done on purpose.
It can also cause strain with your relationship with the GM. Maybe not long term, but in the game. I've seen more than a few good friends not invited to the next game because the GM didn't want to deal with having to hold their hand all the time. Remember what I said about the third or fourth time of having something explained to you? That can get very frustrating for the GM who is also trying to keep track of the rest of the game, and now has to regularly point out to you that you get a bonus here, a minus there, and don't even have to roll over there. I can tell you from experience it gets distracting, and if you enjoy the game and enjoy spending evenings gaming I am sure you don't want to be uninvited to future sessions because of it, right?
The near worst case scenario in causing this strain is that it can seem like you are just not paying attention because you don't care about the game. Now, not only are you slowing things down, but people will start to wonder why you are there, and possibly worse things as well. I know personally, my view is that if I can remember your bonuses/penalties without having ever seen your sheet, than you should be able to. If you can't, my brain just defaults to you don't care/want to be there.
So please, pretty please even, try to know your sheet. The slowing down of the game is fairly marginal to the frustration and potentially real arguments it generates over whether or not you are paying attention or even care about a game that some around the table have put a lot of work into. If it is a brand new game, sure there is leeway. If it is a brand new system, even more. But by the sixth session at the least you should know mostly how your character works.
As a final note, I know that some people have a hard time getting these things down. New systems are scary, sheets are complex, and keeping track of a wide array of bonuses can be hard to do. It happens, there are people who have real problems with it. If you are one of those people, find someone at the game you can trust with seeing your sheet and ask them for help. Having another player do it keeps it from distracting the GM, and can very quickly emulate 'knowing your sheet' as the other person can point out your bonuses and penalties. Most people at the table, in my experience, will be happy to do so. Especially since you knowing your bonuses increases the groups chance to succeed. So don't be afraid to ask for help, ask for clarification on rules, or just for someone to go over your sheet with you and help you make a cheat sheet with all the things you need to be aware of by order of occurrence.
Honestly, I still agree with this post. Sure, it may be a bit ranty and ravy at times, but the information is solid. As a player, I think the chief responsibility I (or any other player) has is to know their sheet. Take some time, look it over, and pay attention to where things are. You don't need to know the system's mechanics to know that your skills are on the front/right of the sheet, and your stats at the top/center; and it goes a lot quicker when you just have to ask "what do I roll?" as opposed to "Where do I find that?"
If I could change anything in this post, I would move up the mention of the people who just can't keep this information straight earlier. There are people who enjoy gaming, are paying attention, but just can't keep system details down. They shouldn't be penalized for that, they should be helped with it. Some people's brains simply work in different ways than others.