Min chef siger, at ansvar og respekt er vigtigere end hurtige svar.

Because chef is a common-gender noun in Danish. With common-gender singular nouns, you use min for my.

• min chef = my boss
• mit hus = my house
• mine venner = my friends

So the possessive agrees with the gender/number of the noun being possessed.

Yes. In Danish, chef usually means boss, manager, or head of something.

That can surprise English speakers, because English chef usually means a cook. In Danish:

• en chef = a boss
• en kok = a cook

So min chef is my boss, not my cook.

Because Danish normally uses a comma before a subordinate clause introduced by at.

Here:

• Min chef siger, at ...

The part after at is a subordinate clause: at ansvar og respekt er vigtigere end hurtige svar.

In English, the comma before that is often omitted. In Danish, it is much more standard to include it.

Here at means that and introduces a clause.

• Min chef siger, at ... = My boss says that ...

It does not mean the infinitive marker to in this sentence. Danish at can have more than one function, so context matters.

Because they are being used as general abstract concepts.

• ansvar = responsibility
• respekt = respect

In Danish, abstract nouns are often used without an article when speaking generally, just as in English:

• Responsibility and respect are more important ...

If you made them definite, it would sound more specific:

• ansvaret = the responsibility
• respekten = the respect

But here the sentence is about the general values.

Because siger is the present tense of at sige (to say).

• at sige = to say
• jeg siger = I say / I am saying
• min chef siger = my boss says

In Danish, present tense verbs usually end in -r.

Because vigtigere is the comparative form of vigtig (important).

• vigtig = important
• vigtigere = more important
• vigtigst = most important

So:

• ansvar og respekt er vigtigere = responsibility and respect are more important

Danish often forms comparatives with -ere, instead of using a separate word like more.

Because end is the word used after a comparative, like English than.

• vigtigere end = more important than
• større end = bigger than
• bedre end = better than

So:

• vigtigere end hurtige svar = more important than quick answers

Because svar is plural here, and adjectives before a plural noun normally take -e.

Compare:

• et hurtigt svar = a quick answer
• hurtige svar = quick answers

So:

• singular neuter: hurtigt svar
• plural: hurtige svar

That -e ending is very common in plural adjective phrases.

Because some Danish nouns have the same form in singular and plural indefinite.

• et svar = an answer
• svar = answers

This is normal for some neuter nouns in Danish. You often tell singular vs. plural from the article, adjective, or context:

• et hurtigt svar = singular
• hurtige svar = plural

The clause after at has normal subordinate-clause word order:

• at ansvar og respekt er vigtigere end hurtige svar

The subject is ansvar og respekt, and the verb is er.

A useful thing to remember is that Danish subordinate clauses often show their special word order most clearly when there is an adverb like ikke:

• Han siger, at det ikke er vigtigt.

Here there is no such adverb, so the clause looks quite similar to English.

Because ansvar og respekt is a coordinated subject: responsibility and respect.

Even though each noun is singular by itself, together they function like a plural idea:

• Ansvar og respekt er ...

In Danish, the verb form er is the same for singular and plural anyway, so you do not see a separate plural verb ending like in some other languages.

Very natural as it is. A Danish speaker could say it exactly this way.

In speech, you may hear the rhythm group the sentence like this:

• Min chef siger,
• at ansvar og respekt
• er vigtigere end hurtige svar.

Also, siger is often pronounced less carefully in everyday speech than the spelling suggests, so learners may not always hear every written sound clearly. That is normal in Danish pronunciation.