It's probably related to pen in the sense of 'to confine'. The anger, for example, can be thought to be confined or 'penned in', like an animal in a pen. So it can't get out until the pressure becomes too great.

Pen can be an instrument that you write with. It also has the meaning of a cage where animals are kept. A pig pen is where a farmer keeps pigs for example. Pent is an Old English conjugation of penned, meaning put in a pen. So, pent up means to be in a cage, to be restricted in movement, to be shut up. He has a lot of pent up anger means that he has suppressed his anger.