Optikeren begynder synsprøven.

Because -en is the definite article attached to the noun.

• optiker = optician
• optikeren = the optician

Danish usually puts the at the end of the noun instead of using a separate word like English does.

In Danish, definite nouns usually take the article as a suffix:

• en optiker = an optician
• optikeren = the optician
• en synsprøve = an eye test
• synsprøven = the eye test

So English uses a separate word, but Danish often builds that meaning into the noun itself.

begynder is the present tense of begynde (to begin / to start).

• at begynde = to begin
• begynder = begins / is beginning

In Danish, the present tense often covers both ideas that English separates into:

• begins
• is beginning

So Optikeren begynder synsprøven can sound natural as either The optician begins the eye test or The optician is beginning the eye test, depending on context.

Because at is only used with the infinitive, like English to.

• at begynde = to begin
• begynder = begins / is beginning

In this sentence, the verb is already conjugated for the present tense, so at is not used.

It is a compound noun:

• syn = sight / vision
• prøve = test

Together:

• synsprøve = eye test / vision test
• synsprøven = the eye test / the vision test

Compound nouns are very common in Danish, just as in German and often in English too.

That is how Danish compound nouns work. The whole compound becomes definite by adding the definite ending to the last element:

• synsprøve = eye test
• synsprøven = the eye test

You do not mark each part separately. The definiteness applies to the whole word.

This sentence uses the basic Danish word order:

Subject + Verb + Object

• Optikeren = subject
• begynder = verb
• synsprøven = object

So it matches normal English word order quite closely:

The optician begins the eye test.

Yes. In this sentence, begynder can often be translated as either begins or starts.

So these are both natural translations:

• The optician begins the eye test.
• The optician starts the eye test.

In everyday English, starts may sound a bit more natural, but begins is perfectly correct.

Usually yes, in ordinary usage. Optiker refers to the eye-care professional you visit for glasses, vision checks, and similar services.

Depending on context, job titles in different countries do not always match perfectly, but for learning this sentence, optician is the right translation.

A rough guide for an English speaker is:

optikeren begynder synsprøven
→ op-TEE-keh-ren beh-GYN-er SYNS-pruh-ven

A few notes:

• y in begynder is not like English y. It is a front rounded vowel that English does not really have.
• ø in synsprøven is also a vowel English lacks.
• Danish pronunciation is often less phonetic than spelling suggests, so it is best to listen to native audio if possible.

They are:

• en optiker = an optician
• en synsprøve = an eye test

And the definite forms are:

• optikeren = the optician
• synsprøven = the eye test

Both nouns here are common gender nouns, which is why they use en and -en.