Is there any way to interpret the set of all functions from a set $X$ to a set $Y$?
There is an interpretation of it as the cartesian product of $X$-many copies of $Y$, but I am asking for a more fun, if you want, interpretation. Maybe something of combinatorial flavour?
If you accept well-ordering, then consider X as a well-ordered set and then the functions from X to Y represent sequences of the elements of Y according to the well-ordering of X.
So, if X is the ordinal numbers then the functions represent sequences of elements of Y extending through the ordinals.
At a more mundane level, if X is specifically the positive integers, then the functions are what we would normally understand as infinite sequences of elements of Y.
Well, that's one interpretation.