Let’s get the easy one out of the way first. To cite is to refer to – you might cite a reference in a document, or use the word to refer to something that has come before, usually a precedent: “to cite precedent, we have let a man into our women’s tennis club before, and so we will have to do so again”.
A site is the place where something is located, whether that’s the piece of land where something is built or the location where something happens. To site something is to place it in the location where it is going to stay or take place.
Sight is the power or faculty of seeing, as well as the more metaphorical extensions like foresight, where you are not physically seeing something, but perceiving something in the future. A sight is something that you see.
When you see the sights, you are, literally, seeing them with your sight. Therefore they are sights, not sites.
But sites can also be sights that a sightseer would view! “The main sights of the trip are the sites of the Pyramids and army encampments”. We see the touristic sights, but they are all sited somewhere.
If it’s a feature, phenomenon or building, etc., that you see, it’s a sight. If it’s the place where that thing is situated, it’s the site of that thing. And if you refer to an essay about the phenomenon, you might cite it!
“Jeremy cited the newspaper article about the argument over the site for the new ‘see the sights of the world’ exhibition in his report on the controversy”.