Negative Past tense
was not [adjective]
Structure
- い-Adjective → short negative past form or polite negative past form
- Casual: おおき
い+ く + なかった → おおきくなかった- Polite: おおき
い+ く + ありませんでした → おおきくありませんでした
- Casual: おいし
い+ く + なかった → おいしくなかった- Polite: おいし
い+ く + ありませんでした → おいしくありませんでした
- Casual: はや
い+ く + なかった → はやくなかった- Polite: はや
い+ く + ありませんでした → はやくありませんでした
Is there a reason the polite version of these adjectives get あり? Why do they get that?
The reason I ask is because うVerbs don’t get that, they just conjugate the stem (I hope that’s the right way to say it) and I can’t find anything online that explains why.
What’s the difference? Or is it just, that’s how it is?
I don’t know the reason, it’s just how the language evolved.
When you say that in order to make a verb polite, one conjugates the stem, what happens is that the auxiliary ます is attached to the verb, and ます has to be attached to the 連用形 (aka masu stem). For adjectives, a way to attach ます directly to the 連用形 (the く form) just never appeared. So that’s why it’s not just 連用形 + ます.
Now where does the あり come from? First, the reason い-adjectives can function like verbs (e.g. as a predicate) is that the い-form includes a “to be” meaning - but the く form does not. But since we can use it adverbially, we can just add a verb to add the “to be” meaning, like ある, and then we can make that verb polite because we know how to make verbs polite. It doesn’t have to be this way, but it’s something we can do.
For example, in order to make 美味しい polite, we can use it as an adverb paired with the polite あります and get 美味しくあります. Like I wrote, it doesn’t have to be this, and there are other polite forms, for example, in modern usage you’ll probably see 美味しいです more often, or the slightly old-fashioned 美味しゅうございます. Of course, we can use the past negative polite form of ある (ありませんでした) to make a past negative polite form 美味しくありませんでした, even though, as the grammar point’s example sentences show, often you’ll see 美味しくなかったです instead. (In both cases, the です variant is much more recent.)
Another aspect of the question is why does ある only appear in these forms. And the answer to that is, it’s not only them. For example, the non-polite かった form developed out of く + あった (the past of ある), the volitional かろう can be analyzed as く + あろう (the volitional form of ある), and so on. If you go back far enough, there’s a ton more in Old Japanese too.
Adding to nekoyama answer, the ありません comes from the negative polite form of the ある.
In other words, with verbs, you simply attach ない to the stem, while for いadjectives you technically attach the negative form of ある.
It all comes way back to classical Japanese, where いAdjectives were called く adjectives. Unlike now, where い of いadjectives conjugates (which is fairly simple) the copula あり (similar to だ and です which are modern copulas, that is words linking predicate to the subject, putting it simply “to be”) had to be added to くadjective so:
Adjective: (寒い cold (weather)
Now: 寒い
Classical Japanese: 寒くあり (isn’t it complicated? Japanese thought the same, so it was replaced with い eventually)
By the way, ある is the attributive form of the same あり (the form used to describe/modify nouns).
So, to make a negative form in the past we had to attach suffix ず
ず attributive (noun modifying form) was ぬ、which likely evolved to modern ない, though there are also other likely etymologies, like evolving from 無し or even both at once. Leaving all that, ず was replaced with ない.
Adjective: (寒い cold (weather)
Now: 寒くない
Past: 寒くあらず
By the way, くあり was often contracted to かり which given birth to other expressions like から (because)
Do you see similarity with the 寒くありません？ 
Verb (死ぬ to die)
Now: 死ない
Classical Japanese: 死なず
So as you can see, the ず was replaced with ない eventually. The verbs never had あり attached to begin with so that’s why we simply have 食べません instead of 食べありません.
But we should get 寒くあらない, not the 寒くない, isn’t it? What happened here?
To put it simply, the whole あらず merged to ない (likely because it was much easier to pronounce).
However, in the polite negative form, the あり still remains, like a fossil, a reminder of the past.
So this is why we have 寒くない and 寒くありません and why negative verb forms do not have ありません but simply ません.
I hope it helps,
Cheers!
Oh that makes a bunch of since thanks for saying it in simple terms so I can understand it 