Judging by the text of the question, the term "egotistical" in the headline was probably meant to be "egoistic." Ayn Rand herself suffered from a similar confusion during the writing of The Fountainhead and made a conscious effort to correct her choice of wording.
Still, the question is ambiguous.
-- Does it seek an explanation of why one's own interests aren't more important to oneself than anyone else's interests?
-- Does it seek an explanation of why one's own interests aren't more important to someone else than the other person's interests are to him?
The questioner might understand the issue of egoism versus altruism a little more clearly if he were to specify to whom in connection with "important." Objectivism points out that all values presuppose an answer to the question: of "value" to whom and for what? This is a key step in Objectivism's validation of man's life as the only objective standard of value for man, and one's own life as the only proper moral purpose of one's own existence. Further explanation can be found in The Ayn Rand Lexicon under the topic of "Values" and related topics such as "Virtue" and "Happiness."
Update: Why Egoism?
What is wrong with being egotistical [egoistic]?
Nothing. It is what man ought to be.
I need a thoughtout explanation of why my interest arent more important than anyone elses....
In other words, the question apparently is asking what the basis for egoism is. The question may also be assuming that egoism means sacrificing others to oneself instead of sacrificing oneself to others, or "doing unto others before they do unto you."
The philosophy of Objectivism provides an extensive development of rational egoism from metaphysics on up, and it's far too much to summarize definitively here. Regarding egoism alone, i.e., selfishness, there are nearly five pages of excerpts on "Selfishness" in The Ayn Rand Lexicon, preceded by nearly another whole page on "Self-Interest." Here is just one small sample, excerpted from VOS Chap. 1:
The Objectivist ethics holds that human good does not require human sacrifices and cannot be achieved by the sacrifice of anyone to anyone. It holds that the rational interests of men do not clash -- that there is no conflict of interests among men who do not desire the unearned, who do not make sacrifices nor accept them, who deal with one another as traders, giving value for value.
Objectivism regards egoism, in essence, as living by productiveness and trade, guided by reason. Does the questioner need a justification for productiveness? But how else are values for man supposed to come into existence, except by someone producing them? Does the questioner need a justification for reason? But how else is man to produce the values his life requires, i.e., how can he know what to do and how to do it? Even if most people merely copy whatever others do, someone somewhere had to originate the knowledge and actions needed to produce values. Does the questioner need a justification for trade? But what better way could there possibly be for man to live in peace, prosperity, and harmony with his fellow man, free of coercive, life-diminishing interference by others?
Objectivism does not, of course, reach egoism "out of the blue." Objectivism digs far deeper, all the way down to basic metaphysics and epistemology, including metaphysical axioms. Objectivism identifies the nature of "value" as such -- values of any kind, rational or not -- recognizing the fundamental nature of all living things as goal-directed entities -- identifying why living things are goal-directed and what the most fundamental, all-encompassing goal is -- identifying how living things act, not only living things in general, but man in particular, man's distinctive capacities and needs -- and, hence, the values that man's life requires, the meaning and role of morality in man's life, why man needs morality and what specific code of moral values is most commensurate with man's metaphysical nature.
Objectivism does not stop at egoism, either. Objectivism goes on to identify the major political implications of the morality of egoism -- namely, individual rights, laissez-faire capitalism, the nature of government and man's need of a proper government that recognizes and upholds man's individual rights.
Egoism in ethics also has implications for esthetics (art) as well as politics and economics.
If the questioner or anyone else has more specific questions about various aspects of egoism and its foundation, by all means let's hear the questions. This is potentially a good website (in my opinion) for discussing them.