Boiled prawns harden in the fridge because the cold temperature causes the proteins in the prawn meat to tighten and contract, expelling moisture and resulting in a tougher texture.
When prawns are boiled, the heat denatures the proteins within their muscle tissue. This process causes the proteins to unfold and then re-bond with each other, creating a network that gives the prawn its cooked texture. However, when these cooked prawns are then refrigerated, the cold temperature further encourages these protein bonds to tighten and contract. This contraction squeezes out any remaining moisture from the prawn meat. The loss of moisture and the increased density of the protein network are what cause the prawns to become hard and rubbery. The longer they are stored in the fridge, the more pronounced this effect becomes. Furthermore, the cold temperature slows down enzymatic activity, but doesn't stop it completely. These enzymes can continue to break down the prawn's tissues over time, contributing to the change in texture.
To minimize hardening, store cooked prawns in an airtight container with a slightly damp paper towel on top to help retain moisture. Consume them as soon as possible after cooking and cooling.