Storage, temperature control, and airflow.

The storage part is the big one, imo. You keep animals in a barn, you don't want to fuck around going to another building for their food, cleaning supplies, etc. It's also more efficient land space wise. Building one or two levels up uses less land, which can then hold pastures, a different barn, whatever.

But it really depends on the type of barn. There's multiple types that gained popularity in the states over time. The oldest styles were English, with simple construction, lots of space for animals, with the storage above, but with usually only two levels. Here in the Appalachians, you see a lot of crib barns, which tend to be shorter on average, and are more for grain storage than animals. There's tobacco barns all over the south that are one or two stories, but the stories are shorter than the ones meant to hold animals and hay.

There's a kind of barn that's specifically built on hills so they have a basement/cellar.

The kind of barns for different types of livestock vary from the ones solely for crop storage as well as each other.

But the reality tall ones are almost always animal barns because you need the height more. You can store grain or whatever on a single level, and just add a little extra height for whatever airflow you need without having an entire extra story.

But when you've got livestock, especially cattle, the storage in the same building becomes mandatory, as does having a lot of extra height for good airflow. You get manure with livestock, and if you don't have good air moving around you end up with sick animals, no matter how well you clean up. There's modern methods that can sorta bypass the manure issues, but they have their own problems.

Even smaller animals like chickens, where you won't necessarily have a second story in full, you'll have a higher roof just for air to move. That's also why the old style longhouse barns work so well. You get a long building with doors at each end, high peaked roofs, with hay storage above. The air moves freely, which cuts down spoilage of fodder, eliminates mildew from built up moisture, keeps the animals healthy (it even reduces disease spread in some cases), and you get the ability to drop hay down instead of having to haul it around.

A three story barn isn't common afaik. It's usually two stories, with a high peaked roof. It ends up being three stories high, but there's only two levels. Some do have an attic built into them, but it's pretty rare since it reduces the benefits. But even that depends on where you are and the type of farm you're on.

You need height if you want to build a structure with a large area and not a ton of internal columns for support.

Also the taller the building is the more potential passive air circulation on hot days.

So you can stack hay.

the barns i played in as a kid stored stuff that needed to be kept dry in the lofts of the barn.

So here's my thoughts as a farmer with 300 head of beef cows:

Old style hip roof barns were like that because before balers, hay had to be put up under cover, and it was put up loose, not baled. You needed these massive lofts to put up any sort of useful amount of winter feed, even if at that time, a large herd was 30 beef and a couple dairy cows. Many of those lofts were used for dances as well if they weren't filled with hay. As bales became common, you didn't need to put as many away in these lofts, and if you did fill them with bales, you'd collapse the loft because they were built for loose hay, not bales.

So bales started be stored outside, either exposed to weather or under a roof cover or tarps. Then round bales came about and they don't really need any sort of cover because they shed water fine because of how they're wrapped.

Modern barns might be 12-14' inside, but that's because they only get used for a couple months of bad weather calving and most cows in areas that are just cold but not rainy make out just fine outside, and would be too hot inside, not to mention spreading diseases. So these barns will often get used for equipment storage and to make them easier to start in winter. The attic will be unused. The extra volume is nice for evening out humidity, but you can also run fans if need be. Many cattle barns are only big enough to run whatever equipment you need to feed and clean it out.

Barns usually have a mezzanine or a second floor for storage. At least the ones I've seen have never been tall but empty.

Dairy-giraffes

Hog barns or farrowing houses typically don't have high roofs and you can accommodate more pigs in the same amount of space.

It’s more of how things were engineered and for what uses. IE if the barn is also used for hay or equipment storage in addition to accommodating livestock

There's more than cows and tractors in barns. Actually, there are infrequently cows in barns; they live in the fields. Barns are typically used to store hay, feed, seed, and tools, such as tractors and their implements, and to provide a dry area for working on equipment and for storing maintenance fluids for that equipment, such as oil and grease.

What you’re seeing is giraffe enclosures. They never let them out because they’re allergic to sunlight, and also they’re illegal. But that super tall roofline is a dead giveaway you’ve got a giraffe operation in your area.