This so-called ink core is called a disposable ink bag

Unscrew the pen and take out the used ink bag. There may be a little bit left in it, just like the milk in the yogurt box. If you can't tear it open and lick it, it's definitely not clean.

Put the new ink bag into the disassembled pen and press it hard. If it doesn’t work, turn it slowly and press it hard. Don’t worry, it won’t be in place before you press it down. It will break, I mean if the pen is not a shoddy product.

Generally, if you press and turn it while you feel it, it will make a "beep" sound, and that's it.

Reference pictures for how to use:

The fountain pen is a widely used writing instrument and was invented in the early 19th century. The pen head is made of metal, and it is smooth and elastic when writing, which is quite smooth. There are obvious trademarks, brands and models on the mouth of the pen cover or on the surface of the pen tip. It is also divided into dip pens, fountain pens, and ink cartridge pens.

Reference: Encyclopedia