Why does the name "epimorphism" refer to a surjective homorphism?

The prefix "epi-" in Greek has several meanings, but a common one is "upon, over". This is similar to the meaning of the prefix "sur-" in French, which was the origin of the term "surjective", introduced by Bourbaki. As such, both give the meaning that the function/morphism "covers" all of its range.

The prefix "epi-" in Greek means "on top of, above". Surjection is a map onto its codomain, and hence the name.

To give another example, the epigraph of a function is the part above the graph.

Also with function between sets sometimes the terms "epic" and "monic" are used instead of "surjective" and "injective"

It comes from the fact that the prefix "epi" is Greek for "upon", "over", or "at". The prefix is also used in, epidemic, epidermis, or epicenter to indicate these meanings. Thus an onto homomorphism is said to be an epimorphism, i.e. a morphism which maps over/upon/onto the range of the function.