Why would an increase in the cost of carry, reduce the value of a put?

It relates to the cost of the underlying. So the cost of carry would be an interest cost. As spending money to buy the asset means thar we lose out on interest. If rates (i.e. CoC) rises you would like to have cash now on deposit rather than the underlying asset to make use of the higher rates. Therefore delaying the sale of the stock with a put is not attractive so the value of a put (a delayed sale of the stock) falls.