An anthem is a poem or a song that celebrates or sings the praises of something. Think of "national anthems," for instance which celebrate a particular nation. "Anthem" also sometimes has a specifically religious connotation, referring to a piece of sacred music or poetry (its original meaning). An anthem is a song of praise to the divine, in other words, often with quotations from a holy scripture.
Ayn Rand's novella is in both respects an "anthem." It's written in a poetic style (Rand herself once classified it as a "poem"), and it definitely sings the praises of something. (What? We'll give you a hint: it starts with "e" and ends with "go.") But Anthem also has the sacred dimension to it, since Rand identifies the individual human self – the ego – as the highest, most holy thing there is. She even goes so far as to call the ego "god" at the end of chapter eleven (11.20-22). And as far as scripture is concerned, many readers find that Rand's style is suggestive of a sacred or mythological text, which is almost certainly deliberate on her part. (See "Writing Style" and "Symbolism, Imagery, Allegory" for more on this.)
We should also mention that Rand herself considered the book to build up to the last two chapters, which are literally an anthem to the ego (Ayn Rand, Anthem (Centennial Edition), New York: Plume Books, 2005, pg. vi). They basically consist of the main character proclaiming its undying glory in the most powerful language he can muster.