I suppose the idea was that if you remove something from one location, it's no longer in that location or even close to it, but instead far away.
And so the past participle entfernt, used as an adjective, has come to mean "far" (often in the combination weit entfernt) as well as the original meaning "removed".
The noun Entfernung is then related to this sense of distance.
Compare also the English noun "remove" which can mean "distance" (e.g. "at some remove").
The noun for "the (act of) removal" is the typical German verbal noun: just use the infinitive as a neuter noun, i.e. das Entfernen "the removing". For example, das Entfernen der Zecke war nicht schmerzhaft "Removing the tick was not painful".
Theoretically, die Entfernung could also mean "the process of removing" but in practice, the conventional meaning "the distance" is so strong that it could end up being confusing.
The word entfernt also has the equivalent removed in English, though it's not nearly as common. Quoting Wiktionary:
removed (comparative more removed, superlative most removed)<pre>
Separated in time, space, or degree. Now that we are here one week removed. Of a different generation, older or younger Steve is my second cousin once removed.</pre>
Abstand is the distance/separation/gap between two items.
Entfernung is how far away something is.
So they both measure something similar; I suppose the difference is how they "look at it".
I suppose Abstand is usually used for closer things, that may be parallel. For example, the user manual to a room heater might tell you to keep an Abstand of at least 30 cm between it and any wall.
Entfernung might be more for further things.