Le panier est déjà plein, car les poivrons et les courgettes sont en promotion.

In French, nouns usually need an article. So le panier means the basket.

Unlike English, French often does not drop the article in ordinary sentences.
So:

• le panier = the basket
• un panier = a basket

Here, le tells us we are talking about a specific basket.

French nouns have grammatical gender, so every noun is either masculine or feminine.
Panier is simply a masculine noun, so it takes masculine forms:

• le panier
• un panier
• plein (not pleine)

There is no special meaning behind the gender here—it is something you usually just have to learn with the noun.

Est is the il/elle/on form of être (to be), and it agrees with le panier:

• le panier est = the basket is

Plein is an adjective meaning full. Since panier is masculine singular, the adjective stays in the masculine singular form:

• masculine singular: plein
• feminine singular: pleine
• masculine plural: pleins
• feminine plural: pleines

So le panier est plein = the basket is full.

Déjà is an adverb meaning already. In this sentence, it modifies the idea is full, so it comes after the verb est and before the adjective:

• Le panier est déjà plein.

This is a very natural word order in French. Compare:

• Il est déjà prêt. = He is already ready.
• La salle est déjà pleine. = The room is already full.

Both can mean because, but they are used a little differently.

• car is often a bit more formal or written
• parce que is very common in everyday speech

In your sentence:

• Le panier est déjà plein, car...
  = The basket is already full, because...

You could also say:

• Le panier est déjà plein parce que les poivrons et les courgettes sont en promotion.

That would sound completely natural too.

The subject is plural because it contains two things joined by et:

• les poivrons = the peppers
• les courgettes = the zucchinis/courgettes

Since the subject is plural, the verb must also be plural:

• singular: est
• plural: sont

So:

• Le panier est plein
• Les poivrons et les courgettes sont en promotion

Because les is the plural definite article for both masculine and feminine nouns.

So:

• singular masculine: le poivron
• singular feminine: la courgette
• plural masculine: les poivrons
• plural feminine: les courgettes

That is why both nouns use les in the sentence.

The -s marks the plural in writing.

• un poivron → des / les poivrons
• une courgette → des / les courgettes

In most cases, that final -s is not pronounced, but it is still important in spelling and grammar.

En promotion means something like:

• on sale
• discounted
• being promoted in a store

So les poivrons et les courgettes sont en promotion means the peppers and courgettes are being sold at a special lower price.

A very common English translation is simply are on sale.

This is a fixed French expression: être en promotion.

Here, en is part of the phrase, just as English uses on in on sale.
You generally learn it as a chunk:

• être en promotion = to be on sale

French often uses prepositions in ways that do not match English exactly, so it is best to memorize the full expression rather than translate word by word.

In standard French, courgette is the normal word.
English varies by region:

• British English often says courgette
• American English usually says zucchini

So les courgettes may be translated as either courgettes or zucchinis, depending on the variety of English you want.

A few pronunciation points:

• Le sounds roughly like luh
• panier sounds like pa-nyay
• est is often pronounced like ay
• déjà sounds like day-zha
• plein has a nasal vowel, roughly plan without a clear n
• poivrons sounds roughly pwav-ron
• courgettes sounds roughly koor-zhet
• sont en creates a liaison: the t in sont is pronounced before en

So in connected speech, sont en promotion sounds approximately like son-tan promotion.

Yes, grammatically you could say:

• Un panier est déjà plein...

But the meaning changes slightly.

• le panier = a specific basket, known in the situation
• un panier = a basket, less specific

In your original sentence, le panier sounds more natural if the speaker is talking about the basket they are currently filling while shopping.

Yes. The adjective plein agrees with panier, which is masculine singular:

• Le panier est plein.

If the noun changed, the adjective would change too:

• La boîte est pleine. = The box is full.
• Les paniers sont pleins. = The baskets are full.
• Les boîtes sont pleines. = The boxes are full.

So this sentence is a useful model for how adjectives match the noun they describe.

Yes. French can say either:

• les poivrons et les courgettes
• les poivrons et courgettes

But repeating the article before each noun is very common and often clearer, especially for learners. Since one noun is masculine and the other feminine, using les before both sounds very natural here.

The original sentence is perfectly standard.