Au magasin, la vendeuse met les vis dans un sachet et le marteau dans un paquet.

Au magasin is a very common way to mean at the store/shop. It uses à + le = au.

• au magasin = at the store / in the context of the store
• dans le magasin = inside the store in a more physical, literal sense

So Au magasin is a natural scene-setting phrase, like saying At the store, ...

Vendeuse means saleswoman or female shop assistant.

It is the feminine form of vendeur.

• un vendeur = a male salesperson
• une vendeuse = a female salesperson

Because the person is female, French uses the feminine noun vendeuse and the feminine article la.

Met is the third-person singular present form of the verb mettre (to put).

The subject is la vendeuse, which is she / the saleswoman, so the verb must match that subject.

Present tense of mettre:

• je mets
• tu mets
• il / elle / on met
• nous mettons
• vous mettez
• ils / elles mettent

So la vendeuse met = the saleswoman puts.

French usually uses an article where English often does not.

So French normally says:

• les vis = the screws
• le marteau = the hammer

Even when English might say something more general like she puts screws into a bag, French often prefers an article. In this sentence, les vis and le marteau sound like the specific screws and hammer involved in the situation.

Here, vis is plural because it comes with les:

• la vis = the screw
• les vis = the screws

A useful thing to know: the singular and plural of vis are spelled the same, so the article tells you which one it is.

These two words refer to different kinds of containers.

• un sachet usually means a small bag, little packet, or pouch
• un paquet usually means a package, parcel, or wrapped pack

So the sentence suggests:

• the screws go into a small bag
• the hammer goes into a package

French often chooses container words quite specifically, just like English does.

Dans means in or inside.

So:

• dans un sachet = in a small bag
• dans un paquet = in a package

The verb mettre dans is a very common pattern in French for saying that someone puts something into something else.

Because mettre takes a direct object.

In this sentence:

• les vis is the thing being put
• le marteau is also a thing being put

So French says:

• mettre quelque chose dans quelque chose
• literally, to put something into something

You do not need de after mettre in this structure.

The comma separates the opening location phrase from the main clause.

• Au magasin sets the scene
• then the sentence tells you what happens

This is similar to English:

• At the store, the saleswoman...

The comma is not absolutely mysterious grammar here; it is mainly punctuation that makes the sentence easier to read.

Yes. The verb mettre is very common and has several meanings depending on context.

It can mean:

• to put
• to place
• to put on clothing
• to spend time
• to turn on in some expressions

But in this sentence, the meaning is clearly the basic physical one: to put.

Because French is coordinating two parallel parts of the sentence:

• met les vis dans un sachet
• et le marteau dans un paquet

The second part leaves out met because it is understood from the first part. This is completely normal.

So the structure is basically:

• she puts the screws in a bag
• and the hammer in a package

This avoids repeating met unnecessarily.

Yes, if the person has already been identified.

For example:

• Au magasin, elle met les vis dans un sachet et le marteau dans un paquet.

That would mean the same thing, but la vendeuse is more explicit. French often keeps the noun when the speaker wants to be clear about who is doing the action.