I take no pleasure in this prediction. I am not a vegan, nor do I have any strong wish to become one. And yet, I think a lot of people will be vegans in the near future for simple reasons of technology, cost, and environmental sustainability.
A popular talking point among vegans is that a calorie of animal product is much more expensive in terms of land, labor and energy resources to produce than a calorie of plant food. And this is true. It's certainly less efficient to grow corn, feed it to a cow, and then eat the cow. Cut out the middle cow and eat the corn directly! But of course, corn (and any other plant) doesn't taste like cow, and to date the rich West has been willing to pay the premium necessary to purchase the flavor and texture of meat.
Several trends however are coming together which I think will move a substantial amount of our consumption away from animal products towards plant products. From there, changes in politics will finish off animal farming as a major industry.
The first trend is merely the limitation of land resources. Already, 26% of the Earth's surface is devoted to grazing, and 1/3 of our arable land is used to grow crops fed to food animals. I don't know what the Earth's sustainable level of beef production is, but seeing how over-grazing is already a severe problem in some areas, it may well be less than the level we have now. Combine this fixed supply with a growing population and we should expect the price of animal products to rise over time.
The second trend is the improving sophistication of "fake" animal products. Muufri is making "animal free" milk that matches real cow milk protein-for-protein and fatty-acid-for-fatty-acid, just from plant sources. Impossible Foods is doing the same for meat, even going so far as to replicate the hemoglobin in blood from plant sources. The promise of both of these companies, and others working in the same field, is to deliver plant-based products which are indistinguishable from their animal-based counterparts, but at lower cost. Already in blind experiments, according to Impossible Food's research, professional chefs are unable to tell the difference between their products and the real thing during preparation and cooking, and customers can't tell the difference either. And the product is improving from there, since they have precise control over the chemistry. This isn't your uncle's tofu burger.
The third trend is the direct manipulation of DNA using technology like CRISPR. In 2013, the biotech startup Pronutria came out of stealth and David Berry (one of their founders) gave a Google Solve For X talk on their technology. Basically, they created a library of single-celled creatures and DNA tools for modifying them to create the amino acids and vitamins necessary for human health from a continuous process algae farm. Pronutria observed that by growing these nutrients directly from algae we can provide all of the non-calorie nutrient needs (proteins and vitamins, excluding starch and fat) of the entire planet from a non-arable patch of land (or calm ocean water) the size of Rhode Island. Obviously the fresh water and energy requirements are also much lower than standard agriculture.
Taken together, the above three trends say that the costs of animal products will rise, the quality and price of their plant-based substitutes are already near-equal, and the energy and resource cost of plant-based substitutes are already lower and destined to fall much further. Over time the competitive bidding for a good steak from the global rich will drive up the cost of fixed-supply "real" meat, while simultaneously scalable biotech alternatives will come to enjoy economies of scale and learning curves, driving down their prices towards the marginal cost of energy and non-arable land (low).
Will animal farming simply go away? Not immediately. There will certainly be many people who don't want to give up their animal-based foods, and will be able to afford to keep eating it. But there's 4 billion people in Asia who want to eat well, and by the end of this century there may be just as many in Africa. There simply isn't enough grazing land on Earth to feed everyone real meat, and given a sufficiently tasty substitute (which may be even more nutritious than the real thing), the world's poor and middle class will probably go for it. Over time, as more people get used to the idea of plant-based meat and dairy products, political acceptance of the known downsides of standard agriculture (the environmental and ethnical issues surrounding animal welfare) will dry up. I can see a future where so many of the voting public becomes disconnected from eating animal-based meat that it acquires a reputation similar to fox hunting - something vaguely cruel and pointless that only eccentric rich people do. Regulations that make it increasingly expensive and rare would follow quickly from that point, in the name of animal welfare or environmental protection.
On the plus side, for those of you currently saddened by the thought of never having a "real" hot pastrami sandwich in the post-meat future, any real limit on Earth's human-population carrying capacity will be expanded out into the indeterminate future by the events described in this post. Malnutrition will be abolished anywhere supply chains and markets are reasonably functional, and we will become infinitely richer in the ultimate resource. So we'll have that going for us, which is nice.