Why do my gums hurt after eating a lot of meat and then flossing?

Need prof cleaning: I suspect you have gums that aren't as healthy as they can be. Healthy gums will not bleed nor be tender after any eating. If you gums are inflammed to start with, then food will get more easily impacted into the gums and when you clean the gums with floss, it will bleed and be tender. Get a good cleaning and exam by a dentist to eliminate advanced periodontal disease (which you might have).

Spaces between teeth: Meat can always finds it way between teeth that may not have tight contacts with their neighbor or areas without teeth. As it packs in between those teeth it puts pressure on the gums below and also can put pressure on the teeth to "move". Flossing it out can aggrevate it a little too but mostly from the packing of the meat. If you have troublesome areas go see a dentist and see what can be done.