Here's an easy analogy.

View morphisms \\(\text{Hom}(-,X)\\) in a category like extension cords with the inputs not plugged into a wall outlet.

\\[
\overset{\text{black cord}}\longrightarrow \text{Lamp}
\\]

Then the fact that pre-composition runs backward is analogous to the fact that if I want to extend my extension cord to get closer to a wall outlet, then cords are composed backwards,

\\[
\overset{\text{beige cord}}\longrightarrow \
\overset{\text{black cord}}\longrightarrow \text{Lamp}
\\]

here, we're using the \\(\text{beige cord}\\) to get the \\(\text{black cord}\\)'s output closer to a desired input. If we can't, we get another cord and (throwing saefty concerns to the wind) we keep daisy chaining on cords until we can get to a wall outlet (or else you're out in a deserted area and it's actually impossible).