how to eat a native-to-North America common persimmon (Diospyros virginiana)
Make sure it's actually ripe. The fruits start out green, then yellow, then orange, then to their final ripe pinkish-orange. Look at the fruit you have. Is it pure orange? Bright orange? It's not ripe. Don't eat it yet. Sit it on the counter or in the sun for a few days and check again. When it's ripe it'll be almost a pinkish color, and extremely squishy and fragile. If you're thinking "oh no it's dissolving in my hands it has to be bad now" you'd be mistaken, that means it's now perfectly edible.
Clean if off if it's not already clean
take off the cap
Now you have several options:
Eat it one bite at a time
or
Squish it between your fingers until the pulp comes out, then eat that
or
mash a bunch of them (minus the caps!!) in a bowl and then just eat the pulp out of the bowl.
or
shove the whole thing in your mouth
The seeds are not edible, so don't swallow them.
Each seed has a little sac around it of more fruit that /is/ edible, you can use your teeth to clean the seed, and then just spit it out. congrats. that seed can now be planted any you can have your very own persimmon tree. Or you can chuck (or plant it) in the woods and help biodiversity that way :)
Most common persimmons I've found so far only have 1 seed in them, but they can have up to six. The smaller the fruit, the more likely there will be only one, or even no seeds at all.
If the other animals beat you to all the fruit and you want to grow some from seed, do not depair--go to the tree, and start looking on the ground around it. You should be able to find tons of seeds that you can simply pick up and plant.
The protective sac eventually goes away with weather and ants and stuff, so most of the seeds you'll find on the ground will be perfectly clean and won't need any processing. If you find some that still have the sack on them, just put them in a separate bag or something and you can clean them off later. Removing the sack by hand is very easy: Take your thumb nail, and push upward with it. The sack should break open, then you can simply pull it off the seed and there you go, perfectly clean seed ready for planting.
Actually you could probably even keep the sac on and plant it that way, it's extra nutrients, after all.
And if you find any fruits that are too squishy or dirty to eat, you can either cover the whole thing with some dirt or plant the whole thing in a container (1 inch deep should work), or mush it up in some water to get the seeds out. Any pulp that's inedible can be added right into the dirt of wherever you're plant the seeds for free fertilizer.