Why must my personal understanding of right and wrong also apply to other people? What if I think something's wrong for me to do, but I don't care if other people do it (e.g. procrastination)?

Because you care about other people, and other people are relevantly similar to yourself. This applies to both instrumentally relevant details, like the character of a person you're going to hire, and more personal concern, like whether your brother is living a good life.