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Wiktionary says that the word погода (weather) has a plural form: погоды.

I see in Google that although the word is usually used in the singular form, the plural form is used either. For example, here is a quote from a book by Chekhov, a classic Russian writer:

Хорошие погоды нынче стоят! (Source)

Literally, the above sentence means, "Good weathers now stand," and such a formulation appears weird to me.

Could you explain the logic and usage of the plural form of погода? In particular, I would like to know the precise meaning of the above phrase and why Chekhov chose the plural form.

Mitsuko, you seem to have developed the talent of stumbling upon 'false quotes' (remember your Bismark question?)

Disclaimer: I realize that this is not an answer to the question about the plural form of погода. Might add something on this later on. I just find it quite interesting that this rather popular citation from Chekhov cannot actually be found in any of his works.

"Хорошие погоды нынче стоят! То ли чаю выпить хочется, то ли удавиться."

Фразой из пьесы «Дядя Ваня» до сих пор подписывают фотографии в
соцсетях, но в измененном виде. Ее превратили в предложение, где автор
якобы говорит о том, что день прекрасный, и можно или выпить кружку
чая, или повеситься.

Такой цитаты у Чехова нет, зато в пьесе один из
героев в диалоге говорит: «В такую погоду хорошо повеситься».

From "Дядя Ваня":

…

Войницкий. Я молчу. Молчу и извиняюсь.
Пауза.

Елена Андреевна. А хорошая сегодня погода... Не жарко...
Пауза.

Войницкий. В такую погоду хорошо повеситься...

Edit:

As to "appears weird to me" - it does not sound particularly weird for a Russian ear. As @YellowSky mentions in the very first sentence, this can quite naturally be understood as the weather observed for a period of several days.

After all people do say "сегодня хорошая погода, а вчера была плохая погода", so here we go - хорошие погоды нынче стоят..

But people don't speak like this these days other than to make a bit of a joke (actually, I use погоды from time to time myself and I have no idea where I picked this up. I think this was long before I had read or seen "The 12 chairs" but that's impossible to verify anyway).

Погоды [стоят] means "the weather during several days". The most famous instance of that 'weather in plural' used in a work of fiction is in Двенадцать стульев ('The Twelve Chairs'), a classic satirical novel about con artists by the Odessan Soviet authors Ilf and Petrov, published in 1928:

Теперь такие погоды замечательные.

In the classic 1971 Soviet film after the novel, the phrase was a bit changed and became

Сейчас такие дивные погоды стоят!

Here is the fragment from the film where it is said: https://www.youtube.com/watch?v=r4nVtiN99H8

Most of the modern usage of погоды (pl.), is just quoting or paraphrasing that film episode, not the book, and surely not Chekhov.

Outside of that phrase, nobody uses погода in plural, as far as I know.

Could you explain the logic and usage of the plural form of погода?

Погода is a regular noun (ж 1а by Zaliznyak's classification), and so it has plural just like any (or should I say many?) other.

As your quote from "Checkhov" turns out to be fake, here is one from Kuprin's "The Garnet Bracelet":

В середине августа, перед рождением молодого месяца, вдруг наступили отвратительные погоды, какие так свойственны северному побережью Чёрного моря.

As you see, "weathers" are just "weathers" and nothing else.