How does a pokemon’s nature affect its stats? Why does a pokemon with a jolly nature get an increase in speed, while its special attack? How about a sassy pokemon getting lowered speed, with an increase in special defense?

Well, that’s an odd way to refer to capabilities. I may have to use that; it’s a far less unwieldy term.

In any case, anonymous, while it might seem odd, a pokémon’s personality has a lot to do with its battle capabilities because of the distribution of will. Let’s take your traits for an example. Let’s assume that you have two pikachu. One of them is a generally jolly pikachu (“generally” because this is what trainers mean by “jolly,” as really, a pokémon’s personality is a complex set of interacting traits, the same as a human’s or animal’s), and the other is a generally sassy pikachu.

Now, where personality comes into play is in a pokémon’s preferences in self-training and development. If you have two human beings, one may prefer to build their strength and stamina, while another may prefer studying to develop themselves intellectually. In the same way, consider the jolly pikachu. Jolly pikachu tend to be highly energetic and perky, and as such, they prefer running about and getting things done quickly, as the constant exercise of speed for them is an adequate outlet for their naturally energetic personalities. Consequently, jolly pikachu tend to be faster than others, simply because they’ve preferred to train themselves to be faster. By contrast, they’re not so adept at using special (or distance) attacks, as they focus more on releasing their energies via running. It isn’t that they never have energy to use special techniques; it’s just that they don’t release as much energy via using techniques as other pikachu do. Eventually, their ability to do so is very much like a bookish human’s ability to lift weights. Even if you attempted to train them to do so, it will take quite a bit of time before they can reach the sorts of levels an athlete may have achieved. Hence, to an outside observer, a jolly pikachu is a fast one, but it’s not particularly a hard hitter.

Meanwhile, there is the sassy pikachu. Unlike jolly pikachu, sassy pikachu tend to be more self-defensive, due to the fact that they are, well, rather cheeky. In order to get away with the mischief they’re inclined to make, sassy pikachu train themselves to be able to handle a few hits. The reason why they focus so much on their special defense (as opposed to physical defense) is because sassy pikachu are usually more intelligent and realize that their species is generally faster than anything they may pester. As such, these pikachu also realize that they can simply run out of arm’s reach, but they can’t outrun an Earthquake. For this reason, they see no point in developing their agility or offensive capabilities; instead, they focus completely on their defensive capabilities.

Any so-called nature follows this same form of logic. While, yes, all pokémon can receive special training to build up certain stats (you can, for example, train a jolly pikachu to launch more powerful distance attacks than other jolly pikachu), by and large, the differences between each nature are the results of pokémon developing skills in their own times, according to what they think would be best for themselves.