Le cordonnier du centre-ville connaît la vendeuse de la quincaillerie.

Because du is the required contraction of de + le.

• le centre-ville = the downtown / the city center
• de + le becomes du

So:

• du centre-ville = de le centre-ville

French always makes this contraction with de + le and de + les:

• de le → du
• de les → des

Because quincaillerie is a feminine singular noun, so its article is la, not le.

That gives you:

• la quincaillerie
• de la quincaillerie

Only de + le and de + les contract.
de + la stays de la.

French uses connaître and savoir differently.

Use connaître for:

• knowing a person
• being familiar with a place
• being acquainted with something

Use savoir for:

• knowing a fact
• knowing information
• knowing how to do something

So here, since the object is a person (la vendeuse), connaît is the right choice.

• Il connaît la vendeuse = he knows the saleswoman
• Il sait son nom = he knows her name
• Il sait réparer des chaussures = he knows how to repair shoes

Usually it suggests some degree of acquaintance or familiarity.

With a person, connaître often means:

• he knows her
• he is acquainted with her
• he is familiar with who she is

It does not automatically mean they are close friends, but it is usually stronger than simply having heard of someone.

Because in this sentence those words are being used as full noun phrases:

• Le cordonnier = the shoemaker
• la vendeuse = the saleswoman

French normally uses articles with nouns much more often than English does.

A useful contrast:

• Il est cordonnier. = He is a shoemaker.
  
  • no article after être when stating someone’s profession
• Le cordonnier connaît la vendeuse. = The shoemaker knows the saleswoman.
  
  • article needed, because the profession word is now the noun itself in the sentence

Because French nouns referring to people often show grammatical gender, and in many cases that matches the person’s sex.

Here:

• cordonnier is masculine
• vendeuse is feminine

So:

• le cordonnier = the male shoemaker
• la vendeuse = the female saleswoman

Related forms:

• la cordonnière = the female shoemaker
• le vendeur = the male salesperson

English usually does not mark this directly in the noun, but French often does.

Each de phrase attaches to the noun right before it.

• Le cordonnier du centre-ville
  • du centre-ville describes le cordonnier
• la vendeuse de la quincaillerie
  • de la quincaillerie describes la vendeuse

So the sentence is built like this:

• [Le cordonnier du centre-ville]
  subject
• connaît
  verb
• [la vendeuse de la quincaillerie]
  object

This is a very common French pattern.

Because these phrases are not mainly telling you where the action happens. They are part of the noun phrases and help identify the people.

• le cordonnier du centre-ville = the downtown shoemaker
• la vendeuse de la quincaillerie = the saleswoman from/at the hardware store

If you used au centre-ville or à la quincaillerie, you would more clearly be expressing location:

• Le cordonnier est au centre-ville.
• La vendeuse est à la quincaillerie.

So de here is doing more of a describing/identifying job than a simple location job.

In this sentence, the most natural English idea is something like:

• the saleswoman from the hardware store
• the saleswoman at the hardware store

French often uses de + noun where English might use:

• from
• at
• a noun used like an adjective, as in hardware-store saleswoman

So the exact English wording can vary a little, but the French structure is perfectly normal.

The circumflex in connaît is mainly a spelling feature of the verb connaître.

A few useful points:

• it helps mark this verb as part of the -aître family
• historically, circumflexes often reflect an older missing s
• in this word, it usually does not create a dramatic pronunciation difference for modern learners

So you should mainly treat it as an important part of the correct spelling:

• il connaît
• not il connait in traditional standard spelling

It is pronounced roughly like ko-NEH.

More exactly:

• con- sounds like French co with a clear n
• -naît sounds like nèh / nɛ

The final t is silent.

So:

• connaît ≈ ko-NEH

Because centre-ville is a fixed compound noun in French. It is normally written with a hyphen in standard spelling.

It functions as one unit meaning:

• downtown
• city center

So:

• le centre-ville
• du centre-ville

You should learn it as a set expression.

It can mean either, depending on context.

Possible meanings include:

• hardware store
• hardware goods / metal household items / ironmongery

In this sentence, because it says la vendeuse de la quincaillerie, the natural meaning is the hardware store.

Yes, the main sentence pattern is very similar:

• subject + verb + object

Here:

• Le cordonnier du centre-ville = subject
• connaît = verb
• la vendeuse de la quincaillerie = object

What may feel different is that French often uses noun + de + noun where English might prefer a shorter compound:

• la vendeuse de la quincaillerie
• English might say the hardware-store saleswoman

So the overall word order is familiar, but the way French builds descriptions is often different.