Why does your pinkie hurt when you hit a part of your elbow?

Uhhh... it doesn't?

There is a nerve that runs along that part of your elbow called the ulnar nerve. It gives sensation to your pinkie finger and a part of your ring finger. It runs along the "backside" of your elbow. So bashing your elbow in that area sometimes compresses your nerve there and causes pain/weird sensations in that area. You're directly smashing/compressing the nerve when that happens.
The sensations to the rest of your fingers are from different nerves. One of the other main ones is called the median nerve. It does run along your arm as well, but in a pathway that is not as exposed at the elbow like your ulnar nerve is. It runs along your "elbow pit" area, so bashing your elbow doesn't affect it at all.
But the median nerve is commonly compressed at the wrist. This happens in carpal tunnel syndrome, where you can get pain/tingling/weird sensations in your remaining fingers.
Other nerves can be compressed in other ways, but the above two are probably among the most common.
Check out this image of what nerve supplies sensations in any given area of the arm/hand:
https://m.instantanatomy.net/diagrams/arm061.png
Here is an image that shows where the main nerves run along the elbow. This view is looking at the "elbow pit" head on. The backside of the elbow is essentially behind this image if that makes sense.
https://lemmy.world/pictrs/image/93de5fe2-a798-435e-beff-fbc42897d839.png
Anatomy stuff is pretty cool, imo. I guess I'm a bit of a dork for stuff like that.
- My answer was going to be that there's probably a nerve there.
  Your answer is much better.
- Fun fact: The ulnar nerve equivalent to carpal tunnel is cubital tunnel.
- Ulnar Nerve AKA Funny Bone

It's a bit like why drilling into a wall might make the lights go out if you hit a cable. Your brain only registers "feeling" in any part of your body because a nerve carried that information to it. The nerves from your lower arm and hand pass your elbow. Hitting the nerve directly causes signals in it which you brain interprets as pain in your fingers. Presumably the nerves for the pinky side of your hand are slightly more exposed.

Because I hit my elbow against my pinkie.
- I keep seeing your username, and I just have to tell you I love it. I have now spent time considering the implications of a sentient loom, and I will continue to do so because they are so vast. Thank you.

When you hit your elbow, you can hit the ulnar nerve. This nerve, among other things, carries tactile information from the pinky and usually part of the ring finger, so hitting it causes an unpleasant sensation in that area.