My perspective on this question changed over the years. Now I consider it to be pretty mature and well educated.
Whether anybody wants to admit it or not, we do care what others think of us. Not only because we, as humans, want to be validated, but because we have to. Some may care less than others, such as myself, but we still do care.
Take for example a case everyone has to (or should) go through. Interviewing for a position at a job. Do you want to interview there and them not to like you? Of course not! Them not liking you means they won't hire you. You want to be well received, you want them to like you. You may not change yourself for them (and I don't suggest you do), but you still want them to like you. And that means you care what they think of you. You may not take it to heart and let that affect you and change you, but you still care. If only for that particular circumstance.
Take another example. You're meeting your girlfriend's folks. You don't know them. They don't know you. Therefore, you don't really care about them as people just yet. The only connection you have to them is the girl you claim to love. You want them to like you. If they don't like you, you'll be heartbroken that you don't have their validation.
Being well liked and well received means you are validated. Humans like to be validated. That does not mean that they require validation from others in order to feel good about themselves. It just means that they like to be validated from a source outside of their own frame of mind. It's human nature.