In order to explain what privacy is, I would suggest to define it as a person’s right to share and to withhold information according to one’s own preference. This does not only include information that people might not want to share with anybody, but also information which people are only happy to share with a small number of people (such as very close friends or family). If you define privacy in this way, it almost answers your next question: protecting your privacy ensures you have the right to decide for yourself which pieces of information about yourself you want to share with other people.
This is why you could argue that people care about protecting privacy, as it gives you a certain amount of freedom and independence. If protection of privacy didn’t exist, everything you think or do could potentially become common knowledge, whether you like it or not. I’m sure you’d agree that most people do not want every detail of their personal life to be known by everybody, particularly with regard to highly personal topics such as sex, emotions, health, and family. By protecting your privacy, you also protect yourself from feeling embarrassed—you don’t have to share embarrassing personal information with others, you can keep it secret from them if you wish and therefore avoid embarrassment. If there was no protection of privacy, then this would no longer be the case, as everything could potentially come out into the open.