To cite something means to do right by whoever said it and give them credit — for instance, if you add a brilliant statement to a paper but you’re not the one who originally wrote it, you should cite, or point to, the original author.
Using cite with the meaning "to reference" or "to note" is just one of several similar uses. A military or other official report can cite someone for an outstanding act of service, and a legal representative can cite individuals, or summon them, to show up in court. To cite a source when writing or verbally repeating something first spoken by someone else is to give a "citation" or "notation" that gives credit where credit is due.