Having raised two boys and a daughter past this stage, let me assure you, this is just a phase. Toddlers feel a certain amount of anxiety even as they are learning to explore on their own. They are old enough to leave your side in a moments notice, and in due time, they will, but for now, they want to be babies in the sense that they want constant attention before the cord it officially cut...
Up until this time, you have showered them with attention, affection and given them all they could ever want, if not more. Now, as they are getting older, you are happy to let them explore their world, within limits of course, and they sense a fair amount of disconnect between you and them. Feeling a loss will happen many many times during their lives, but this is their first time and they do not have the vocabulary to properly express it. The same thing will happen when the first girlfriend/boyfriend thing is over. They will feel like the world is going to end, but we all know it does not.
So how do you manage it for the next few months? As a father, we react differently to toddlers when they exhibit these types of behaviors than mothers do. When the kid falls out or is constantly whining, we ignore them or immediate correct the behavior in a way father's do, very sternly. The reason why we react this way is because we know what you know: There is nothing physically wrong with our kid. They are just attempting to get their way. If you submit to their will now, good luck in breaking the cycle later on. This is when you establish who is who in the relationship. So when your child is crying for no reason at all, ignore the self-serving behavior initially, correct it by establishing that you are not going to change your behavior to submit to their wishes and, if needed, sit your child in a quiet corner until they settle down. Repeat as often as necessary until they learn you will not tolerate that type of behavior.
Understand that your child is exceptionally smart. They are learning new things all the time, even how to communicate their wants, needs and desires without words. But they do not understand yet that it is not your sole job in life to cater to their every need. There are some things they must do on their own. Learning to "just be" is one of those things you must teach them now. From this lesson, they will develop a measure of self-sufficiency and self-confidence, an important step in getting started on the next step: Potty training.
Honestly, if you are a mother to a son, you will have a much harder time during this phase. Little girls seem to get through this phase in a month or less (but then again, I'm a Dad and little girls are more attached to their father than their mother). My two boys are much closer in age, so the older one helped break the younger one as he was going through the cord cutting phase.
Look, your child is so much smarter than you give them credit for. No, they cannot pass the SAT right now and I'm not talking about book smarts. They are developing critical thinking and survival skills. They are learning how you react to what they do and your child knows exactly how to get your attention. They are using that knowledge to their advantage in getting what they want from you. They know how effective the simple word "No" is in getting you to change how and what you do in reference to them. These skills are extremely important to them because they do have words to express what they want to say.
Respect their intellect but set the foundation now for how your relationship will progress in the future. The winner of the battle of the terrible twos usually dictates what happens in the relationship moving forward. If you don't correct their behavior now, when they turn four, you will be asking how do you keep your child from hitting you or wondering what happened when your 9 year old is telling you to shut up. Start early, be consistent, be firm and be fair because it will be too late when they are standing there looking you eye-to-eye and your only option is to call the police.