I've come across this problem a few times.
Could be debris/grit in either the top of the filler (bit on the left) or at the bottom of the flusher mechanism.
This is what I did.
Turn water off, flush the loo.
In your first photo, you can see the cap (on the left hand side 'Filler') under which is a silicon diaphragm, this may contain debris that stops the valve shutting off the water. You might be able to prise the cap off or twist anti clockwise, hold the bottom section still though.
If you can see the diaphragm is clean stick it back together, if not, flush with a cup of water.
Turn water on, but get an assistant to make sure the cap doesn't ping off... or put the lid on the cistern.
That might fix, if not, check the flusher mechanism (chrome button on top), it should be able to be removed, usually anti clockwise twist.
Look for old loo blocks/crud blocking the diaphragm in this until, if it's blocked it will let water into the pan constantly.
Sorry it's a bit long!
good luck.