Vinteren er ikke den årstid, jeg bedst kan lide, men december er hyggelig på grund af jul.

Vinteren is the definite form of vinter.

• vinter = winter
• vinteren = the winter

In Danish, you often use the definite form when talking about a season in a general, familiar way, especially as the topic of a sentence:

• Vinteren er lang.
• Sommeren er min favorit.

So Vinteren er ikke ... is very natural.

In Danish, months, seasons, and holidays are normally not capitalized.

So:

• vinter
• december
• jul

The only reason Vinteren has a capital letter here is that it is the first word of the sentence.

Both forms can exist, but den årstid is very natural when the noun is followed by a relative clause like jeg bedst kan lide.

So:

• den årstid, jeg bedst kan lide = the season that I like best

Here, den is the definite article for a common-gender noun, and årstid is a common-gender noun.

You may also see årstiden, jeg bedst kan lide, but den årstid is a very common and clear choice in this kind of structure.

Because Danish often omits the relative word som when it would be the object of the clause.

So both of these are possible:

• den årstid, jeg bedst kan lide
• den årstid, som jeg bedst kan lide

Both mean the same thing. The version without som is very common in everyday Danish.

Because this is a subordinate clause (a relative clause), not a main clause.

In Danish:

• main clause: adverbs usually come after the finite verb
  
  • Jeg kan bedst lide vinteren.
• subordinate clause: adverbs usually come before the finite verb
  
  • ... årstid, jeg bedst kan lide

So the word order changes because of the clause type.

Here bedst means best or most of all.

It is the superlative form used with kan lide to talk about preference:

• Jeg kan bedst lide sommeren. = I like summer best.
• Hvad kan du bedst lide? = What do you like best?

Danish normally uses bedst with kan lide, not mest, when talking about which thing you prefer.

Kan lide is a fixed Danish expression meaning to like.

Examples:

• Jeg kan lide kaffe. = I like coffee.
• Hun kan lide musik. = She likes music.

Even though kan often means can, in this expression kan lide functions together as one idea: like.

Also notice that after modal verbs like kan, the next verb stays in the infinitive without at:

• kan lide
• not kan at lide

Because the adjective agrees with what it describes.

Here it describes december, and in this sentence the adjective appears in the basic form:

• december er hyggelig

Compare:

• Det er hyggeligt. = It is cozy.
  Here hyggeligt is used because det is neuter.
• En aften er hyggelig.
• Et hjem er hyggeligt.

So hyggelig is the correct form here.

Hyggelig is one of those Danish words that does not match perfectly with just one English word.

Depending on context, it can mean:

• cozy
• pleasant
• warm
• comfortable
• friendly
• nice in a relaxed, inviting way

In this sentence, it suggests that December feels cozy and pleasant, especially because of Christmas.

Yes — it is best to learn på grund af as a fixed expression.

It means:

• because of
• due to

Examples:

• på grund af regn = because of rain
• på grund af sygdom = because of illness
• på grund af jul = because of Christmas

So in this sentence, it explains the reason December feels hyggelig.

Here jul is used in a general sense, meaning Christmas as a holiday or season in general.

So:

• på grund af jul = because of Christmas

Using julen would sound more specific, like the Christmas season or that particular Christmas.

In many expressions like this, Danish prefers the bare noun when speaking generally.

Because jeg bedst kan lide is a relative clause describing den årstid.

So the comma marks off the extra clause:

• den årstid, jeg bedst kan lide

Danish often uses commas in places that feel more explicit than in English, especially around subordinate clauses.

Because men joins two main clauses:

1. Vinteren er ikke den årstid, jeg bedst kan lide
2. december er hyggelig på grund af jul

In Danish, it is normal to put a comma before men when it connects two full clauses.

Because men is a coordinating conjunction, and it does not cause inversion by itself.

So after men, normal main-clause word order stays:

• men december er hyggelig

You get inversion in Danish after certain fronted elements, for example:

• I december er det hyggeligt.
• Derfor er december hyggelig.

But men alone does not trigger that change.