The prefix ab-, in some verbs, can denote abnormality or wrongness, not unlike English mis-: e.g. abutor "misuse, abuse". This seems to be the sense of ab-orior, something like "be misborn" (since orior can be used as a synonym for nascor). L&S actually list aborior in their entry for ab -- scroll down to III.A describing the "mis-" sense.
The original sense of aborior was probably "die", with "be miscarried/aborted" and "miscarry/abort" as derived sense.
Orior originally meant "rise" but took on meanings of "appear" and "be born".
The prefix ab- is used here to mark absence or deprivation.
I think aborior is comparable to the way that French uses disparaitre (disappear) euphemistically to mean "die".
I use wiktionary for etymology every now and then, and am by no means an expert.
Aborior is said to be made from ab and orior:
ab (from, away from; outside of; at a distance; completely, thoroughly; absence of; more remote)
orior (rise, get up; appear; become visible; born; come to exist; originate)
TKR mentions abuse. This may have to do with pushing the person you are abusing away from you emotionally. To add top that, you also have words like abandon and abduct. You also have words like abdomen (away from the mouth), and abdutor muscles (away from the midline). Ab- by itself doesn't necessarily have a negative connotation. With abortion, you are sending the fetus away from you. Away from your abdomen and away from life; the fetus is being taken outside of you.
A few terms descended from orior mean to attack. This is probably where terms like "rise up" came to be. "Birth of a nation" and stuff like that. With abortion, you are rising above what life would be like with a burden attached. The fetus is appearing and becoming visible. It is kind of weird that abortion is descended from a word meaning birth, but you are essentially forcing the birth of a fetus that can not live without you. This can be seen as an attack on the fetus. The fetus will rise to heaven if you believe in that.
That alone should clear up a lot, but we can dig deeper. The root of ab has theoretical descendants in several languages with pretty much the same meaning. I am not sure what this means.
The root of orior means (to move, to stir; to rise, to spring; to quarrel, fight). The root's descendants also invoke a similar meaning, but are more varied. Some of the descendants mean boil, and run. The words seem to be used in a context of high energy. In the context of abortion, you save lots of energy aborting a fetus. You are also using energy to end a life.
I hope this helped with you understanding. It is always fun exploring the wikitionary.