I’m not an egoist anarchist, but the other answer to your question is wrong and I don’t want it to be standing there without correction. Egoism in an anarchist context goes back to “The Ego and His own” by Max Stirner, a German philosopher. He was not an anarchist, but has been and is still very influential on the movement.

Here’s his works on the anarchist library: https://theanarchistlibrary.org/category/author/max-stirner

I haven’t read it myself yet, but read some short summaries. Afaik he wants everyone to acknowledge that we’re basically working in our own self-interest all of the time BUT that this requires cooperation and that helping others without immediate material benefit can be part of our self-interest, e.g. because it makes us feel good or because it builds relationships that might be beneficial later.

Egoism comes from one of two sources, so you can pick your poison. Ayn Rand is popular with libertarians, conservatives, and other short-sighted, smooth-brain ideologies. Left Hand Path Satanism is the other source. That form of egoism isn’t inherently ignorant, but it attracts edgelords and misanthropes.

Egoism works in two philosophical aspects. Psychological and ethical/moral (there is a difference between ethical and moral but here I’m gonna use them interchangeably for simplicity) Egoism as a psychological stance argues that one is self-interest and everything one does is for that end. Moral/ethical egoism argues that one should act in their own self-interest. Egoists like myself use egoism in the political sense to argue that our institutions, religions, concepts of good and bad are merely “spooks of the mind.” Simply put they are fixed concepts that only exist to hinder and impede our true nature and we ought to remove them from our behavior and thought process. Egoism in this interpreration therefore allows us to see it as compatible with anarchism because it also believes hierarchies such as the “spooks” are unjustified and should be dismantled.