I am under the impression that "ergibt" means "to give" or "results," in the sense of logical implication. I have seen it both by itself and with the word "sich." I understand that "ergibt sich" means "giving one," in the sense that one now knows this result. But I do not understand what kind of situation calls for using "sich" versus not using "sich." Could someone please clearify this.
It depends on whether the result, in german das Ergebnis, is the object or the subject of the sentence.
If the sentence has the form "A ergibt B", then the result, here B, is the object.
Zwei plus drei ergibt fünf.
Two plus three results in/ amounts to/ yields five.
If the sentence has the form "A ergibt sich aus B", then the result, now A, is the subject.
Fünf ergibt sich aus zwei plus drei.
Five results from two plus three.
I try to explain it with an example:
 Die Beobachtung von Vögeln ergibt interessante Erkenntnisse.
 Aus der Beobachtung von Vögeln ergeben sich interessante Erkenntnisse.
In Sentence  »Die Beobachtung von Vögeln« is the subject and »interessante Erkenntnisse is the object«. In Sentence  »interessante Erkenntnisse« is the subject and the object.
You're wrong about "ergibt sich". The "sich" part does not mean "one", which would probably refer to the reader, but it means something like "results itself", which in German occasionally is the way to form what in English would be an intransitive verb. Compare:
I opened the door.
Ich öffnete die Tür.
The door opened.
Die Tür öffnete sich.
The same happens with "ergeben", as other answers have pointed out already:
Propositions A and B yield Proposition C.
Sätze A und B ergeben Satz C.
Satz C ergibt sich aus Sätzen A und B.
Proposition C follows from Propositions A and B.