Augnlæknirinn sagði að ég þyrfti sjónpróf áður en ég keypti ný gleraugu.

Because -inn is the suffixed definite article, meaning the.

So:

• augnlæknir = eye doctor
• augnlæknirinn = the eye doctor

Icelandic usually adds the definite article to the end of the noun instead of putting a separate word before it.

Also, Augnlæknirinn is the subject of the sentence, so it is in the nominative singular.

Not exactly. Augnlæknir is closer to eye doctor or ophthalmologist than to optician.

• augnlæknir = a medical eye specialist
• an optician is more the person who helps fit or sell glasses

In everyday translation, people may sometimes simplify it as eye doctor.

Here að means that.

It introduces a content clause after sagði:

• sagði að... = said that...

This is not the same að as the infinitive marker in phrases like að fara = to go. Icelandic uses the same word for both functions, so you have to tell from context.

Because this is a subordinate clause, and subordinate clauses in Icelandic do not follow the normal main-clause verb-second pattern in the same way.

So after að, the usual order is:

• conjunction + subject + verb

That gives:

• að ég þyrfti = that I needed / would need

This is very common in Icelandic subordinate clauses.

Þyrfti is the past subjunctive form of þurfa.

This form is commonly used in indirect or reported speech after a past reporting verb like sagði.

Compare:

• direct speech: Þú þarft sjónpróf. = You need an eye test.
• reported speech: Hann sagði að ég þyrfti sjónpróf. = He said that I needed / would need an eye test.

A useful comparison:

• þarf = need / needs (present)
• þurfti = needed (past indicative)
• þyrfti = often would need or a reported/less direct needed

So þyrfti is a very natural choice here.

Because each clause needs its own subject.

The sentence has more than one clause:

• að ég þyrfti sjónpróf = that I would need an eye test
• áður en ég keypti ný gleraugu = before I bought new glasses

Even though both clauses have the same subject, Icelandic still states ég in each clause.

Because Icelandic has no indefinite article.

So:

• sjónpróf can mean an eye test or eye test, depending on context

If the noun were definite, Icelandic would usually add the article to the end:

• sjónprófið = the eye test

Here the noun is indefinite, so the bare form sjónpróf is exactly what you would expect.

Sjónpróf is the direct object of þyrfti, so it is in the accusative singular.

However, sjónpróf is a neuter noun, and many neuter nouns have the same form in nominative and accusative singular. So even though it is accusative here, it looks unchanged.

Also, sjónpróf is a compound:

• sjón = sight, vision
• próf = test, exam

So the whole word means vision test or eye test.

Because gleraugu is normally a plural-only noun in Icelandic, just like glasses in English.

So Icelandic treats it as plural:

• gleraugu = glasses

That is why the adjective is also plural:

• ný gleraugu = new glasses

You would not normally use a singular form here to mean a pair of glasses.

Because the adjective has to agree with gleraugu.

Here gleraugu is:

• neuter
• plural
• indefinite
• accusative

For the adjective nýr in the strong declension, the neuter plural nominative/accusative form is ný.

So:

• ný gleraugu = new glasses

This is a very important Icelandic pattern: adjectives must agree with the noun in gender, number, case, and definiteness.

It is the direct object of keypti, so it is accusative plural.

But because gleraugu is neuter plural, the nominative and accusative forms are the same, so you do not see a visible ending change here.

So grammatically it is accusative, even though the form looks the same as the dictionary form.

Áður en means before and introduces a subordinate clause.

So:

• áður en ég keypti ný gleraugu = before I bought new glasses

It works a lot like English before when followed by a full clause.

You can think of the structure as:

• main clause
• then a time clause introduced by áður en

Because the whole sentence is framed from a past point of view: the eye doctor said something.

From that past viewpoint, the buying of the glasses is referred to in relation to that time clause introduced by áður en.

Also, with a verb like kaupa, the form keypti can match more than one grammatical category in practice, so the written form does not always tell you everything by itself.

For a learner, the important point is this:

• keypti is the natural form here after áður en
• the clause means before I bought new glasses or, in smoother English depending on context, before buying new glasses

Yes. That is a very reasonable literal translation.

A few English versions could work, depending on style:

• The eye doctor said that I would need an eye test before I bought new glasses.
• The eye doctor said that I needed an eye test before buying new glasses.
• The eye doctor said I should get an eye test before buying new glasses.

The Icelandic grammar does not map perfectly onto just one English wording, so more than one natural English translation is possible.