How does stronger dollar drive down prices of commodities and/or materials like oil?
There are two reasons I can think of for this effect.
The more economic reason is that commodity prices are usually pro-cyclical. In economic booms those commodity producing countries and their currencies do quite well (their currency goes up and hence USD goes down). And vice versa, in uncertain economic times, there is not as much demand for commodities, and hence less demand for currencies of commodity producing countries (USD goes up and price of commodity declines).
Another reason is more due to unit of account. Many of these commodities are traded in dollars even though the end users are global. If USD went up 1% against all currencies, however the main suppliers and end users of a commodity (say copper) are mostly in Chile and China respectively, then conceivably the cost of copper denominated in Chilean Peso might not have changed at all. This could happen because the Chilean supplier would be willing to sell their copper at 1% lower in USD terms and still get the same profit in Chilean Pesos terms. Therefore, in certain cases it is just a matter of perspective.