I will speak only about the Objectivist (Ayn Rand) version of ethical egoism.
Some key points are:
There are no conflicts of interest between rational men. (Think longterm and big picture.)
No one should ever sacrifice himself. Altruism is bad.
There is no benefit or need for sacrifice, because there are no conflicts of interest between rational men.
If you do what's in your best interest, it will overall be most beneficial to the world, to society, etc. There is no choice between one or the other. No conflict.
It's important that I take responsibility for my own life, and you take responsibility for yours, rather than vice versa. I have better control over my life, I'm in a better position to deal with the problems in my life, I have more incentive to solve my problems, etc. And, further, I own myself and should have freedom to make my own decisions. Since I have the final say about my life, you can't be responsible for my life (since, ultimately, it's out of your hands), I have to be.
From self-ownership, other rights follow such as property rights (since I own my labor and what it creates).
You can help other people on a voluntary, consensual basis, as long as it isn't a sacrifice to yourself. There are many reasons that helping others would be in your own interest and not a sacrifice, such as enjoying the activity, learning something from doing it, a reasonable expectation of return favors in the future, or helping someone who is a value to you (e.g. if you value your wife, you'd want to help her in various ways so she's still available to interact with in the future for maximal benefit – meaning she's still alive, healthy, happy, etc. in a simple and extreme case, by helping her stay alive, you get the value of future time with her. as long as the gained value exceeds the price of helping her, you win.)
the view that there is a harmony of mankinds interests is closely related to the arguments for capitalism, (classical) liberalism, freedom, and that kind of thing. it's really important here. i can elaborate on this if you have questions (if you do, please give some indication of what you already know about capitalism, liberalism, Rand, etc, if anything).