Cette promotion sur les concombres est meilleure que celle d'hier.

Because promotion is a feminine singular noun in French.

French demonstratives agree with the noun:

• ce = masculine singular
• cet = masculine singular before a vowel sound
• cette = feminine singular
• ces = plural

So:

• ce livre = this book
• cet hôtel = this hotel
• cette promotion = this sale / this special offer

Not exactly. In this sentence, promotion means a special offer, sale, or discounted deal in a store.

So une promotion sur les concombres means something like:

• a sale on cucumbers
• a special offer on cucumbers

It is a false friend in some contexts, because English promotion often means an advance at work or advertising/publicity. French promotion can sometimes have related meanings in other contexts, but in shopping language it very often means a special deal.

In French, une promotion sur... is a common way to say a sale on... or a special offer on....

So:

• une promotion sur les pommes = a sale on apples
• une promotion sur le café = a sale on coffee

Here, sur does not literally mean physical on. It is just the standard preposition used in this retail expression.

Because the sentence is talking about cucumbers as the item/category affected by the sale, not just some cucumbers in a vague sense.

After expressions like une promotion sur..., French often uses the definite article:

• sur les concombres
• sur les tomates
• sur le fromage

This is very common when speaking generally about a product category in a store.

Because bon has an irregular comparative form in French.

Just as English says better instead of more good, French normally says:

• bon / bonne = good
• meilleur / meilleure = better

Since promotion is feminine singular, the adjective must agree:

• un prix meilleur would sound wrong here anyway
• une promotion meilleure -> more naturally une meilleure promotion
• after est, we get est meilleure

So meilleure is the feminine singular form of meilleur.

Because it describes promotion, which is feminine.

In French, adjectives usually agree with the noun they describe in gender and number:

• masculine singular: meilleur
• feminine singular: meilleure
• masculine plural: meilleurs
• feminine plural: meilleures

Here:

• Cette promotion = feminine singular
• so: est meilleure

Celle is a demonstrative pronoun. It stands for that one and replaces the repeated noun promotion.

Instead of saying:

• Cette promotion sur les concombres est meilleure que la promotion d'hier

French very naturally says:

• Cette promotion sur les concombres est meilleure que celle d'hier

So celle means that one, with promotion understood.

Because promotion is feminine singular, the pronoun is celle.

Related forms:

• celui = masculine singular
• celle = feminine singular
• ceux = masculine plural
• celles = feminine plural

Yes, that is grammatically correct:

• Cette promotion sur les concombres est meilleure que la promotion d'hier.

But que celle d'hier sounds more natural because it avoids repeating promotion.

French often uses celui / celle / ceux / celles to avoid repetition when the noun is already clear.

Because de contracts before a vowel sound.

Hier begins with a vowel sound, so:

• de + hier -> d'hier

This kind of contraction is very common in French:

• de + accord -> d'accord
• de + habitude -> d'habitude

So celle d'hier means the one from yesterday.

Yes, in this sentence it works very much like yesterday's in English.

• celle d'hier = the one from yesterday / yesterday's one

French often uses de where English may use an apostrophe:

• le journal d'hier = yesterday's newspaper
• la réunion d'hier = yesterday's meeting
• celle d'hier = yesterday's one

Because que is the standard word for than in comparisons.

Examples:

• plus grand que = bigger than
• moins cher que = less expensive than
• meilleur que = better than

So:

• est meilleure que celle d'hier = is better than yesterday's one

Yes. The structure is very standard:

• Cette promotion sur les concombres = subject
• est = verb
• meilleure = adjective/complement
• que celle d'hier = comparison phrase

Literally, the structure is:

• This sale on cucumbers is better than the one from yesterday.

Everything is in a very natural order for French.

Yes, but that would make a different sentence structure.

You could say:

• Cette promotion sur les concombres est meilleure que celle d'hier.

Or:

• Cette promotion est une meilleure offre que celle d'hier.

And in other contexts, French often puts meilleur / meilleure before the noun:

• une meilleure promotion
• un meilleur prix

But in your sentence, meilleure comes after est because it is part of the predicate: is better.

A careful approximate pronunciation is:

Set pro-mo-syon sur lay kon-kombr eh may-yeur kuh sell dee-air.

A few useful notes:

• cette sounds like set
• promotion ends with a nasal sound, not a fully pronounced n
• les sounds like lay
• est is pronounced eh
• meilleure has a y-like sound in the middle: may-yeur
• celle d'hier links smoothly, and d'hier sounds roughly like dee-air

In normal speech, French flows together quite a lot, so listening to native audio is especially helpful for this sentence.