Xjyuk

In Firefly, a space western, they use horses on planets in space that are definitely not Earth.

Is there any canon explanation as to when the horses got sent through space, how they are transported, and why they use horses in space?¹

¹_{other than "it's supposed to be a friggin' space western"}

The show is always very vague when it comes to any details of the exodus. I cannot find a direct reference in canon but frozen embryos and other genetic material is the logical answer and is one that I have often seen used in other sci-fi stories.

As for "why", that is not specifically explained in the show either but the answer to that is also "frozen embryos." A frozen embryo is significantly smaller, lighter, and easier to transport than any piece of machinery that can perform any of the roles filled by a horse. Tractors, Jeeps, etc. are big and heavy plus they need fuel and spare parts. Equipment needs "infrastructure." Horses only need water, grass, and oxygen.

Remember that it is only the poorer outer rim planets that use horses. You never see them on the inner planets and don't see many horses in any larger cities of the rim. Those outer rim planets are "terraformed just enough to make them habitable" before having settlers dumped on them. Under those more primitive conditions animals are a better option.

The Serenity RPG offers a fairly high level overview of the Exodus. In short, when it became clear that Earth-that-was could no longer support human life, gigantic generation ships (known as Arks) carried large numbers of people to what became the central planets of Londinium and Sihnon. After a short period those with an urge to move away from the Central Planets did so, taking with them some of the animals that had been carried in the Arks along with the human settlers.

Even after continued refinement, the process of terraforming a moon or
a planet takes decades. Terraforming requires atmospheric processing
plants, the regulation of gravity, environmental adaptation and the
introduction of creatures great and small brought from Earth-That-Was—
everything from algae and bacteria to insects, birds, and mammals. The
power to make such jing chai changes is astonishing, but is not
without its limits. While most all terraformed worlds are suitable for
human life, each has its own quirks.

As to why they use horses, this is also addressed in the same source

When you’re scratching food out of moon rock without many credits in
the bank (if there is a bank), you tend to figure out what’s important
pretty quick. Why waste the money on a power-driven tool when an
ordinary hammer will do the job? Why worry that your fancy grav-car
will break down on you and you can’t get the spare parts, when you can
raise your own horses?

The Firefly RPG Corebook addresses the ubiquity of horses on the outer worlds.

[On the Border planets] Technology is around some, but it’s a lot scarcer and harder to fix.
Even broken Cortex datapads or a train’s console can spell disaster if
’n you rely on that tech to get by. Truth be told, that’s why horses
are the most common (and cheapest) form of transportation.

How? In spaceships.

It's already canon in this show that cows can easily be transported between planets by spaceship:

We can assume it would be similarly easy to transport horses to outer-rim planets. Just as on Earth they can be carted around in horseboxes, in space they could be taken between planets in some kind of "space horseboxes".

Why? Because it's a friggin' space western.

I know, I know, you're asking about the in-universe explanation, and that was intended as an out-of-universe one. However, it also works equally well as an in-universe explanation! A "western" is something set in the Wild West, a place untamed and on the rugged edge of civilisation. That's basically what the outer-rim planets are too.

People would use horses there for the same reason as they used them in real history: a quick, convenient way of getting around, in a culture where technology hasn't dominated (or has emerged only in certain ways: spaceships, holographic pool balls, and so on). Besides, we don't see much in the way of roads. When you're travelling through the wastelands, a horse makes more sense than a car (or even a jeep) anyway.

We do see one guy who favours a fancy car over a horse:

But all of his fancy tech is pretty useless in the end against a guy on a horse with his fists. On the outer-rim planets, life is tough and so are the people. It just fits the whole atmosphere and environment to use horses instead of some more advanced means of transport, for getting between places by land.

See also: Why don't the outer planets have advanced technology?