This is fairly straightforward. All individuals should try to keep in mind increasing their own personal happiness and fulfillment. This principle means that each individual should be important to that individual. Every individual is entitled to try to make himself or herself happier and more fulfilled.
Universal Morality applies to all advanced species in the universe. However, for simplicity and clarity, I will assume that anyone reading this is of the human beings species.
This principle does not differentiate which humans we should care about. It doesn't say that we should care more about humans the closer they are to us, physically or genetically. It doesn't say we should care more about humans that live in our neighborhood, our city, our state, or our country. It doesn't say we should care less about humans who live in a distant part of the world. It is a very prowerful concept that every human should have a concern for every other human. It leads away from competition, selfishness, and war. The opposite concept would be that the closer someone is to us, genetically or physically, the more we should care about them. Although this is the idea most people instintively believe, it is the concept that leads to tribalism and physical conflict, all the way from urban gangs to world wars. The idea that we should all try to care about everyone will lead toward the end of conflicts and wars.
This means we should be trying to not sacrifice the future for the sake of the present. All people are equally important, including people who haven't been born yet.
If someone is trying to hurt you, you should not necessarily attempt to maximize his or her happiness. The more people are trying to follow these principles, the more everyone should be trying to care about them. This type of mutual caring is the best hope for the future of our world.
Sentient means capable of feeling or perception, or being conscious. On earth, in general that means animals, as opposed to plants. This principle means we should care about the happiness and fulfillment of all animals. If we find other sentient (conscious) beings in the univers, we should care about their happiness and fulfillment also.
Universal Morality applies to all sentient beings in the universe. That means that other advanced beings should care about all of us. To the extent they care about us, they are moral. My presumption is that any civilization that has not adopted a moral code similar to UM will have destroyed itself before it has developed the technology to travel far enough to find us. It may be that all civilizations do destroy themselves when they develop technology advanced enough to do so, and never do adopt these principles. It may be that there are advanced civilizations who have adopted these principles who are waiting for us to adopt them also before they contact us.
To the degree other creatures have the potential to follow this principle, and try to follow it, we should care about them. This means that there may be creatures who do have the potential to follow these principles, but choose not to. It is not immoral to care less about the happiness and fulfillment of these creatures.
This is how we grade how much we care. We should care about other creatures in proportion to their levels of consciousness. Should we care the same about a fly as a dog? No, because a fly has much less ablility to feel happiness or fulfillment (or lack of pain). We don't need to care about the happiness or fulfillment of bacteria (assuming they have some degree of perception) because they have no ability to feel happiness or fulfillment. (Obviously we could never survive if we were concerned about the lives of bacteria, since our immune sytems are killing them all the time.) However, we should care about all animals on earth, and the more advanced (conscious) they are, the more we should care about them. This means we should care about the happiness and fulfillment of mammals, birds, and even fish. Probably we should care more about the happiness of mammals than of fish, because mammals have more ability to feel happiness or fulfillment, and have more ability to feel pain. However, we should still care some about fish.