In fact, there are four channels through which the real interest rate affects real commodity prices (aside from whatever effect it has via the level of economic activity).
First, high interest rates reduce the price of storable commodities by increasing the incentive for extraction today rather than tomorrow, thereby boosting the pace at which oil is pumped, gold is mined, or forests are logged.
Second, high rates also decrease firms’ desire to carry inventories (think of oil held in tanks).
Third, portfolio managers respond to a rise in interest rates by shifting out of commodity contracts (which are now an “asset class”) and into treasury bills.
Finally, high interest rates strengthen the domestic currency, thereby reducing the price of internationally traded commodities in domestic terms (even if the price has not fallen in foreign-currency terms).