Strictly speaking, it wouldn't have killed every dinosaur, but rather all of the biggest ones, with the smaller ones adapting to climate change and becoming modern birds.

The idea is that the impact of the meteor kicked up so much dust, it would have dimmed the sun and made earth much cooler for a period of time. We already observe this happening on a smaller and shorter scale with major volcano eruptions. As the earth cools and dims, large animals have trouble finding enough vegetation or prey to stay fed and warm, and mass extinctions occur. Smaller animals don't need as much bulk food to survive, and there would have been much more plant matter available once the biggest dinosaurs that used to eat by the ton had starved.