Relevance of change in Inventory in calculation of Free Cash Flow

Free cash flow (FCF) can be obtained by $$\text{FCF}=(\text{Revenues}-\text{Cost}-\text{Depreciation})(1-\tau)+\text{Depreciation}-\text{Capital Expenditures}-\Delta(\text{Net Working Capital})$$ where $$\tau$$ is the corporate tax rate and $$\Delta(\text{Net Working Capital})=\Delta(\text{Cash}+\text{Inventory}+\text{Receivables}-\text{Payables})$$ is the increase in net working capital over the period of interest. Besides inventory, all of these items appear to be monetary, except for depreciation which only affects the FCF through taxation, and so that contribution is also monetary.

But what about inventory? A change in inventory does not generate any monetary gains or losses, or does it? If it does not, how come a change in inventory is included in the calculation of the FCF?

When you manufacture a product, you incur some production cost. However, you do not account for this production cost in the $$\text{Cost}$$ line of the income statement until you actually sell the product. While the product is sitting in your inventory waiting to be sold, you have already incurred some (actual, monetary) cost that is not visible in the $$\text{Cost}$$ line. That is why you have to adjust the reported $$\text{Cost}$$ by $$\Delta\text{Inventory}$$ to obtain the actual, monetary cost for the period.
This is why $$\Delta\text{Inventory}$$ belongs in the FCF calculation.