I don't think people will stop eating meat (at least, not on Earth - space missions will probably be all-vegan until colonies on other planets are developed), though I suspect in 100 years' time, it will typically be eaten as an occasional treat rather than something expected every day.
Part of the reason for that is that more health benefits for eating fish, particularly oily fish, get discovered each year (the same is true for eating vegetables, so expect vegetarianism and veganism to become more popular with time regardless).
Another is that vegetarianism, let alone veganism, is tricky to balance nutritionally for a child. People are apt to eat similar diets as adults as they do as children, which will slow down the rate at which meat exits diets - and the forms that replace it (for example, expect battery-bred chicken to give way to a mixture of free-range chicken, lamb (which is difficult to breed in a "battery" fashion", fish and vegetable-based alternatives).
Finally, meat has traditionally been seen as a rich person's status symbol - some meats still are. (Certainly, 400 years ago, British working-class people were vegetarian for most of the year, because they couldn't afford meat once it was out of season). If meat production drops, especially the cheaper methods most under pressure, that situation will recur.
For myself, I am open to eating meat, although for a number of reasons* I avoid non-offal beef when other options are available, and I've never eaten veal or foie gras. I eat between 2 and 6 portions of oily fish a week (it'll be the higher end this week because I have a cold), but I also enjoy eating lots and lots of vegetables. Even the occasional soya portion or Quorn fillet.
* - The reasons, for anyone interested, are:
Cows have been scientifically demonstrated to have particular friends within their herd. This is not the case with sheep, pigs or (to the best of my knowledge) any other animal that is offered for sale as food in my local shops - at least, not consistently across the lifespan of a typical animal grown for food.
Beef is by far the most polluting type of meat there is. Pretty much every other option (even veal) can, under certain circumstances, either be less ecologically damaging than the vegetarian alternative or complement it (for example, by providing sufficient rich fertiliser to allow more plants to be grown than are consumed by the animals involved, which offsets the artificial fertiliser otherwise needed). This is not the case for beef, because by the time a cow has finished processing grass, there isn't much in the way of nutrients left in its manure. (I'm OK with eating beef offal because, given how much less popular beef offal is than beef steaks, consuming offal creates no more pollution than throwing it away).
Finally, I find it very difficult to cook right. Granted that this isn't really an issue when I'm eating away from home, but surely it's better for me to pick food I'm less likely to waste by burning or turning into gristle?