Why is Riley sad?

Riley isn't sad, she's numb. This is shown by her personality islands shutting down and breaking apart. This isn't just sadness; it's an absence of any emotion at all. Her core memories are gone, and Anger, Disgust, and Fear stop trying to control Riley's emotions after their disastrous stint at trying to react the way Joy would.

Everything that makes Riley who she is gets lost, and her real-world response is to just shut everything out.

(Incidentally, this is -- to some -- an accurate depiction of depression: you lose the ability to feel either joy or sadness, you're prone to fits of anger or despair, and eventually you just stop feeling anything at all.)

I completely agree with Roger's answer, but I'd like to add something. He used the word "numb"; I'd like to add one more word, "apathetic", that is, she feels nothing. Not sadness: nothing. She doesn't care about anything anymore. Showing the difference between having no feelings and being sad isn't simple, but the movie found a way: at a certain point, when Anger, Disgust and Fear have failed to behave appropriately, Riley simply doesn't react to their commands. No matter what they do, it doesn't affect her. It looks like their console is broken. This is apathy. But when Sadness comes back and can act, Riley feels something at last, and bursts into tears. This is the clear difference: before that moment, she couldn't even cry, because you need to feel sad to cry, and she didn't.

And this is the message of the movie: while both states (apathy and sadness) are bad, sadness isn't entirely bad, it is something that is actually useful in life, something that can help you. Joy understands it when she sees the memory in which Riley was very happy, surrounded by all her team mates, and realizes that this happened because some moments earlier she was clearly, visibly, evidently sad (not apathetic!). Her sadness is what made it clear to the others that she needed some comfort, and it happened. And, on the bus, her sadness is what caused her to get off and go back to her parents, which was obviously the right choice: so, again, sadness helped her.

At the beginning of the movie, everybody does their best to prevent Sadness from doing anything, because they genuinely think it's the best course of action. At the end of the movie, they have all understood that sadness isn't completely negative or useless: while unpleasant, it is necessary. Growing up isn't always pleasant, and trying to exclude sadness from one's life simply can't work. Instead it must be accepted and faced. When you learn to do it, you are stronger and you can start to enjoy your life again. But if you don't go through this process, and you simply remove sadness from your life, then you'll end up being apathetic, which is worse.

So, without getting into the discussion about depression (I am no psychologist, I can't help there), I'd say that Riley doesn't get sad. She is not happy either, true, and that's why her state can be confusing; but her problem is that she is not sad, and therefore she can't experience the positive effect of sadness.