A word-for-word translation in English could be: “from the vicinity of Swann's place”. That's not a perfect translation: the range of du côté de is broader than the vicinity, it also includes the place itself. “From a small area centered on Swann's place” is closer to the right meaning.
Coming on to your question proper, the preposition chez admits an unusual construction (in addition to normal preposition-like use, as in “Je vais chez Swann”). In this construction, chez behaves somewhat like an article, but it is an article that stacks: it must be followed by a complete noun group which may itself include an article: “chez Swann”, “chez lui”, “chez le boulanger”, “chez mon cousin” are all valid noun groups. This noun group may in turn be preceded by a preposition.
In “du côté de chez Swann”, the noun group “chez Swann” is a complement of the noun “côté” introduced by the preposition “de”.
In the Trésor de la langue française, this construction is described by the second bullet point under A.1.a:
En raison du sens « dans la maison de », le groupe prépositionnel peut être lui-même précédé d'une autre préposition à valeur locale, le concept « dans » étant alors neutralisé et chez signifiant « la maison où habite..., séjourne habituellement... »
“Du côté de” is not a preposition, but it is a group of words acting like a preposition. Note that chez can only be used in this way when it is in a context that invites a location. For example, “I'm coming from his home” can be translated as “Je viens de chez lui”, but “I visited his home” cannot be translated as “* J'ai visité chez lui”, you need to translate the concept of “home” explicitly, e.g. by “J'ai visité sa maison”. (The idiomatic translation for “I visited him” would be “Je lui ai rendu visite”, “J'ai visité sa maison” implies exploring the house itself.)