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).