Heat olive oil in a large skillet over medium heat; cook and stir onion and garlic until browned and fragrant, 3 to 4 minutes. Add collard greens and red pepper flakes; cook and stir until greens are slightly wilted, 2 to 3 minutes.
Combine vegetable broth, cider vinegar, brown sugar, liquid smoke flavoring, and paprika in a large pot; add collard greens mixture. Cover pot and simmer until greens are very tender, 40 to 50 minutes. Strain collard greens before serving.