Inventory Cost of the stock means the costs associated with holding a particular stock. So, if any market maker holds any position in any of the stock, there are some inherent costs associated with it. The major cost is the interest cost or opportunity cost of capital. There were other costs like Storage costs related to physical shares etc. but all these are now not much an issue due to electronic form. So, the major inventory cost associated with holding a stock is the interest cost.

Now, if this inventory cost increases, the market makers would be discouraged to hold more stocks with them or if they were to hold more and incurr more inventory costs, that they would like to get it covered. The only source of covering this cost is through bid ask spread which is their source of income.

So, if this cost increases, the market maker needs to widen the bid ask spread to cover that costs. So, "An Increase in the inventory cost of a particular stock will Widen the Bid - Ask Spread".