Relationship between the Tragedy of the Commons and negative externalities?

The reason why it boosts supply is that the costs are not born by produces. For example, since you can’t own fresh air due to lack of property rights (tragedy of the commons) firms don’t need to pay for using up the clean air by pollution and hence it does not enter their cost functions.

Recall decrease in costs shifts supply right as at lower cost firms are willing to supply more at any price.

I created a graphical example of this for you in overleaf. As the graph show in presence of externality the supply $S$ shifts right and new supply is $S_E$. Demand remains unchanged because the externality affects production costs not preferences or other factors affecting demand. The new equilibrium will be at the intersection of $D$ and $S_E$. In this new equilibrium price is too low and quantity too high compared to social optimum which would be at the intersection of demand and supply without the externality.

The vertical distance between the supplies is the value of the external costs (externality).