Why is it so important for the Project manager to carefully manage stakeholders?

Your stakeholders are those who have a vested interest (stake) in whether your project succeeds or fails. As a project manager, actively managing the successful delivery of your project cannot be done without carefully managing the project-to-stakeholder relationship. In turn, managing this relationship can’t possibly be done well if you don’t know who your stakeholders are. So, let’s ask who your project’s stakeholders are to begin the important process of stakeholder identification.

Because stakehlders expectations are their definition of the success of the project and you certainly aim that see it a success.