Learn Hardcore Russian: В этой корзине самые свежие помидоры и огурцы. - In this basket are the freshest tomatoes and cucumbers.

в этой корзине is in the prepositional case (singular, feminine). In Russian you use the prepositional case with в or на when talking about location—answering “Где?” (Where?). So в корзине literally means “in the basket,” and этой is the prepositional form of эта (this).

Why is there no verb есть (to be/there is) in the sentence?

In Russian the present‐tense form of быть (“to be”) is almost always omitted. You could say В этой корзине есть самые свежие помидоры и огурцы, but in everyday speech you normally drop есть, leaving just В этой корзине самые свежие помидоры и огурцы.

How do you form the superlative degree, as in самые свежие?

The regular way to make a superlative in Russian is самый + adjective in positive form, agreeing in gender, number and case with the noun:

Masculine singular: самый свежий
Feminine singular: самая свежая
Neuter singular: самое свежее
Plural (all genders): самые свежие
(There is also an old‐fashioned suffix -айший that yields свежайший, but it’s quite literary or poetic.)

Why do самые and свежие have different plural endings (-ые vs. -ие)?

Russian adjectives split into two declension types:

Adjectives with a “hard” stem like сам- take -ые in the plural (самый → самые).
Adjectives with a “soft” stem or ending in –ж, –ш, –щ like свеж- take -ие in the plural ( свежий → свежие).
Both still agree with the nouns in case, number and gender.

Are помидоры and огурцы in the nominative or accusative case here?

They are in the nominative plural, functioning as the (implied) subject of the sentence. Note that for inanimate masculine nouns the nominative and accusative plural forms look identical, so you can’t tell by ending alone.

How would you say the same idea if there were just one tomato and one cucumber in the basket?

You’d switch to masculine singular superlatives. For example:
В этой корзине самый свежий помидор и самый свежий огурец.
Since both помидор and огурец are masculine singular, you could also say В этой корзине самый свежий помидор и огурец, but repeating самый свежий makes it clearer you mean both items.