The vast majority of food plants (as opposed to herbs, which are also edible but eaten in small quantities) require full sun for cultivation. It follows that you must therefore clear land in order to grow them. This is particularly true for grains and legumes.
If you depend on plants for your protein then that means you must always clear land to obtain your protein. You also need to find arable land on which to grow that protein. With a population of six billion and only so much arable land, that becomes problematic. This quite aside from the detrimental health effects of eating so much grain and legume, with so many people forgetting the traditional methods of preparing such foods.
Animals do not require cleared land. Cattle can graze under light tree cover, pigs can forage under slightly heavier tree cover, and chickens don't care where you put them as long as there is something to eat. Where humans have found it advantageous to raise animals on cleared land it is so they can look out for predators and keep an eye on the herd or flock, not because the animals require full sun.
Animals also do not require arable land. The land can be fertile enough for grass or forest but does not have to be suited to crops. Therefore animals can be raised in a wider variety of areas than crops are.
This has implications for both deforestation and, as an extension, climate change. In the sense that pumping oil out of the ground for fuel adds carbon to the carbon cycle, it's true enough that petroleum has the potential to contribute to climate change. But removing forest also contributes to climate change because (1) you have fewer trees removing carbon from the air and sequestering it, and (2) fewer trees means higher local temperature. Witness what has occurred in Iraq. Did you know Iraq was once covered by cedar forest, so thick in some areas that sunlight never touched the ground? True story. It was all wiped out for cities and farms. I would imagine, given their latitude, that they had something more like a temperate climate before that happened. If you have ever lived in a house that was partly shaded by trees and partly not, you've noticed this effect too. Or if you've lived in a city and observed that it was hotter in the city than it was in the surrounding countryside.
To add to the petroleum problem, we've gone from using animal labor and biological inputs (what folks call "organic" now--I like the French biologique better, it's more accurate) to using petroleum fuel and petroleum/natural gas inputs. That's not sustainable, it adds to the carbon in the atmosphere and it sucks fertility right out of the ground. People think making plants healthy is all about the NPK. Not so. Soil comes from rotten biological waste as well as from minerals and if you leave part of that out you wind up with less nutritious produce. And refusing to use animals in agriculture also means you have to import exotic plant bits even if you are using "organic methods"--which again participates in the petroleum economy. The more we depend on the latter, the more drilling accidents we're going to see and the more excuses we will make to ourselves for not transitioning away from it as a fuel. You could get everybody to stop driving gasoline cars tomorrow and they'd still panic at the thought of raising food without Big Oil. Until we transition away from that and get back to using animals again, it's going to continue to be an issue. (We'd solve the Big Ag problem too. You can't run a huge industrial farm by plowing with oxen--but you sure can run a smallholding that way.)
And here's the thing about using animals in farming. It's dangerous to have more than one breeding male in a herd. It's annoying (and dangerous to the birds) to have more than one rooster in a flock. Plus, even with one male per animal type, farm animals have babies. What do you do with the surplus? They live for several years and they breed at least once a year (multiple times for chickens); you'll run out of space. You only have so many neighbors to accept your surplus. Then what?
Of course the most sustainable thing to do would be to let the forests come back, and the forest animals with them, and just go hunting. But if we insist on maintaining some sort of sedentary lifestyle, meaning we settle rather than living as nomads, then some degree of farming or ranching will be necessary.
And the best farms are more or less semi-closed ecosystems, with plants and animals sustaining one another and providing meals for you, too.