Au fur et à mesure que décembre approche, les magasins se remplissent de sapins et de guirlandes.

It means as, as... progressively, or gradually as.

In this sentence, Au fur et à mesure que décembre approche means something like:

As December gets closer, little by little...

A very important point: au fur et à mesure que is a fixed expression. It is best learned as a whole chunk rather than translated word by word.

Not really. For most learners, it is much more useful to memorize the whole expression.

The word fur is old-fashioned and does not normally appear by itself in modern everyday French. So this is not a phrase you should try to build from scratch. Just treat au fur et à mesure que as one idiomatic connector meaning as / progressively as.

Because it is followed by a full clause:

décembre approche

That clause has its own subject and verb, so French uses que.

A useful pattern is:

• au fur et à mesure que + clause
• à mesure que + clause

For example:

• Au fur et à mesure que l’hiver arrive...
• À mesure que le temps passe...

If what follows is a noun instead of a full clause, French often uses de instead.

Because month names in French usually appear without an article when used normally.

So French says:

• Décembre approche
• Janvier arrive
• Nous sommes en décembre

This is similar to how English treats month names as proper names.

You could also say le mois de décembre approche, but that is longer and a bit more explicit.

Because the subject is décembre, which is singular.

So the verb approcher is in the third person singular present:

• j’approche
• tu approches
• il / elle / on approche

Here:

décembre approche = December is approaching

Because French often uses the definite article to talk about things in a general, broad way.

So les magasins here means shops/stores in general, especially the shops you see at that time of year.

English often uses no article or a different article in this kind of sentence, but French commonly uses le / la / les for general statements.

So:

• Les magasins se remplissent...
  means
• Stores fill up... or The stores fill up...

The verb here is se remplir, which means to fill up or to become filled.

So:

• Les magasins se remplissent de sapins
  means
• The stores fill up with Christmas trees

The reflexive form is used because the subject is not actively filling something else; instead, it is becoming full.

Compare:

• On remplit les magasins de décorations = People fill the stores with decorations
• Les magasins se remplissent de décorations = The stores fill up with decorations

So se remplir is a very natural choice here.

Because se remplir normally takes de.

The standard construction is:

• se remplir de + noun

So:

• se remplir de monde = to fill up with people
• se remplir d’eau = to fill with water
• se remplir de sapins et de guirlandes = to fill up with trees and garlands

You might sometimes see avec in other contexts, but with se remplir, de is the normal and expected preposition.

Because after de, the indefinite or partitive article often disappears.

So instead of:

• de des sapins

French reduces it to:

• de sapins

That is why you get:

• se remplissent de sapins et de guirlandes

This is very common in French grammar.

Compare:

• beaucoup de livres
• plein de fleurs
• se couvrir de neige
• se remplir de clients

In this context:

• sapins = fir trees, but very often understood as Christmas trees
• guirlandes = garlands, festive decorations, sometimes tinsel or even decorative strings/lights depending on context

So the sentence is creating a holiday-season image: stores gradually fill with Christmas trees and decorations as December gets closer.

Because French often uses the present tense for:

• general truths
• repeated seasonal events
• vivid description

This sentence is describing something that typically happens every year as December approaches.

So the present tense works very naturally:

• décembre approche
• les magasins se remplissent

English does the same thing quite often:

• As December approaches, stores fill up...

Yes. You could also say:

Les magasins se remplissent de sapins et de guirlandes au fur et à mesure que décembre approche.

That means the same thing.

Putting Au fur et à mesure que décembre approche at the beginning gives a smoother lead-in and emphasizes the gradual change over time. Putting it at the end keeps the main focus first on les magasins. Both are correct.