There's two sides to the question. Or rather, a distinction that needs to be made. The first is, as others have said, that people fear dying because it can be painful, unpleasant, etc. Or you don't actually die and wind up crippled or brain damaged. That's reasonable and easy to understand.

The other side, the fear of death, is an existential fear. What do you do in the face of annihilation? Personal, intimate, annihilation of your very being?