The engine revs up when I engage clutch to change gears

Whenever I engage my clutch to change gears, my car revs up even though I have released the accelerator.

However, if I remove my foot from the accelerator a second before engaging the clutch (engine braking), it does not happen. Is this what rev hang is? How do I fix it?

I never had this problem and it only started immediately after replacing a snapped clutch cable.

Also, a problem that has been around for longer which could have contributed to this is that when engine braking, my car makes a horrible grinding noise like gears grinding together.