Place tomatoes and their juices in a large bowl. Using your hands, crush tomatoes, breaking them up into small pieces; set aside.
Heat oil in a large skillet over medium-high heat. Add garlic slices and cook until sizzling, about 1 1/2 minutes. Push garlic to one side of skillet and add red-pepper flakes; toast for 30 seconds. Stir garlic slices and continue to cook until lightly golden.
Add tomatoes to skillet and stir to combine. Add hot water to rinse bowl containing tomatoes and add to skillet. Increase heat and stir in salt. Add basil sprigs to sauce, pressing down to completely cover.
Bring to a boil. Cover, and immediately reduce heat to a simmer; cook for 10 minutes. Uncover and continue cooking 5 minutes more. Remove basil from sauce and discard. Let sauce simmer until ready to serve with pasta.