Persimmons are lovely!
Check the exactly variety first, as this dictates what ripeness/how to eat. With the heart-shaped hachiya variety (which looks like the one you have), the best way to eat them is to allow them to be totally ripe (almost overripe).
When you buy them, they tend to be unripe and hard. I recommend putting them in a pot with an apple or two, with the lid on, for 4 - 7 days. The ethylene natural gas from the apple with speed up the ripening process. Check after day 4 to see how the fruit is going.
With this variety, if you eat it raw it will leave a horrible starchy, drying feel in your mouth (like banana peel).
When ready, they should be a dark orange in colour, and be squishy to the touch - if you poke/squeeze it, it should feel like it is about to burst.
You should then be able to pull out the stem part at the top and then rip the fruit apart with your hands. The flesh will be soft and squishy (almost like a jelly!), and you can scoop it out with a spoon (like a kiwi fruit) or dive in face first to eat it!
Enjoy!