The word cycle itself is simple enough when used to mean something that returns to its beginning, something that loops. But how did it also gain the meaning of a literary cycle, defined as a collection of stories with the same set of characters? There doesn't seem to be any obvious link between the two meanings, and nothing I can find gives an etymology of the latter meaning in particular.
The Oxford English Dictionary has the following definition for "cycle":
A series of poems or prose romances, collected round or relating to a central event or epoch of mythic history and forming a continuous narrative; as the Arthurian cycle. Also transf.
Originally used in the Epic cycle [Greek ὁ (ἐπικὸς) κύκλος], the series of epic poems written by later poets (Cyclic poets) to complete Homer, and presenting (with the Iliad and Odyssey) a continuous history of the Trojan war and of all the heroes engaged in it.
I found some more information about this in the 1911 Encyclopædia Britannica:
The expression “epic cycle” in the sense of a poetical collection does not occur before the Christian era; the word κύκλος (“cycle,” “circle”) is used of a special kind of short poem and also of a prose abstract of mythological history; the adjective has the general sense of “hackneyed,” “conventional,” and is applied contemptuously (by Callimachus and Horace) to a particular Alexandrian school of poetry.
Beyond that, one source says that "[t]heir reason for using the term 'cyclic' (if in fact they did) remains obscure", so I'm not sure there is a definite answer, ultimately.
This sense of the word in English dates at least as far back as 1835:
They..formed the basis or nucleus of the epic cycle.
A History of Greece, Volume 1
The adjective form dates back to a1822:
They are the episodes of that cyclic poem written by Time upon the memories of men.