The Gang of Four book says the intent is:
Use sharing to support large numbers of fine-grained objects efficiently.
If you read more you go on to realize that the purpose is save memory by sharing state. Wikipedia is more direct:
A flyweight is an object that minimizes memory usage by sharing as much data as possible with other similar objects; it is a way to use objects in large numbers when a simple repeated representation would use an unacceptable amount of memory
Clojure collections (lists, maps, vectors, etc...) use structural sharing. Lets just see it in action before we even do a light dive:
create a vector a:
(def a [1 2])
Create another vector b using ag
(def b (conj a 3))
Realize that bs 0th and 1st element are shared with a.
(= a (subvec b 0 2))
This is possible because the collection is immutable. This blog post goes into great detail about structural sharing in clojure, and i highly recommend it.
But this picture should give you an intuition. The high level idea is that the pink, orange and blue vector all share some elements and their is a tree structure that efficiently stores and remembers this.
If you need to mutate that state so it changes between a and b then take a look at clojure state mechanisms.
Clojure uses immutable data structure's which makes sharing a feature that you get out of the box. If you need mutability
then their is a rich core library. Finally, if you want really wanted was caching on a function, then look at memoize.