Bugün kayısı aldım; kayısı olgun olunca tadı daha güzel oluyor.

Because kayısı is an indefinite direct object here.

In Turkish:

• kayısı aldım = I bought apricots / some apricot
• kayısıyı aldım = I bought the apricot(s) / that specific apricot

So the lack of the accusative ending -yı tells you the object is non-specific or indefinite.

It can be understood either way from context.

A bare noun in Turkish often works like:

• a/an
• some
• a general category

So:

• Bugün kayısı aldım usually means I bought some apricots
• kayısı olgun olunca... is more generic: when apricot/apricots get ripe...

If you wanted to clearly say apricots as a plural set, you could use kayısılar, but Turkish often leaves nouns unmarked when the meaning is already clear.

Because Turkish often repeats the noun where English would use a pronoun.

English prefers:

• I bought apricots today; when they ripen...

Turkish very naturally says:

• Bugün kayısı aldım; kayısı olgun olunca...

This keeps the topic clear and sounds natural. Turkish does use pronouns, but usually only when needed for emphasis or contrast.

aldım breaks down as:

• al- = buy / take
• -dı- = past tense
• -m = I

So aldım means I bought.

This is the regular past tense used for a completed action.

• olgun = ripe
• ol- = become / be
• -unca = when / once

So olgun olunca means something like:

• when it becomes ripe
• when it is ripe
• once it ripens

Turkish often uses adjective + olmak to express entering or reaching a state:

• hasta oldu = he/she became ill
• hazır olunca = when it is ready
• olgun olunca = when it becomes ripe / when ripe

It makes a time clause meaning when, once, or sometimes whenever.

Examples:

• gelince = when he/she comes
• bitince = when it finishes
• olunca = when it becomes / when it is

The form changes by vowel harmony:

• -ınca
• -ince
• -unca
• -ünce

Here it is olunca because that is the vowel-harmony form that fits ol-.

Because tadı means its taste.

The base noun is:

• tat = taste

With the 3rd-person possessive ending:

• tat + -(s)ı → tadı

So:

• tadı güzel = its taste is nice
• tadı daha güzel oluyor = its taste becomes nicer / tastes better

Also, the t changes to d before the vowel-initial suffix. This is a common sound change in Turkish:

• kitap → kitabı
• ağaç → ağacı
• tat → tadı

Because Turkish usually shows possession with an ending on the possessed noun.

So instead of saying something literally like its taste, Turkish often just uses:

• tadı = its taste
• rengi = its color
• kokusu = its smell

The possessor is understood from context. In this sentence, the thing whose taste is being talked about is the kayısı.

A more explicit version could be:

• Kayısının tadı daha güzel oluyor.

That means essentially the same thing, but it spells out the possessor more clearly.

Both are possible, but they feel slightly different.

• daha güzel olur = a more neutral general statement: it tastes better
• daha güzel oluyor = very natural in conversation; it can suggest a developing result or an observed tendency: it gets/tastes better

Turkish often uses the present continuous -yor for things that are generally true in experience, especially in everyday speech.

So oluyor here sounds very natural.

daha means more.

So:

• güzel = nice / good / beautiful
• daha güzel = nicer / better

With food, güzel often means something like good or tasty, not just beautiful.

So tadı daha güzel oluyor is naturally understood as it tastes better.

Because in Turkish, adjectives come before the idea they describe, and the structure here is really:

• kayısı = apricot
• olgun = ripe
• olunca = when it becomes/is

So the sense is: when the apricot is/becomes ripe.

Turkish does not need a separate word like English to be in every present-tense adjective sentence, but when you add -ınca to make a clause like when it becomes/is..., you attach it to ol-.

Not completely. Turkish word order is flexible, though verbs usually come at the end of their clause.

This sentence has a very natural order:

• Bugün kayısı aldım
• kayısı olgun olunca tadı daha güzel oluyor

You could move bugün around for emphasis, but the version given is neutral and common.

Turkish often uses word order to highlight information, but the final verb position is still a strong pattern.

The semicolon links two closely related clauses:

• first: I bought apricots today
• second: a related comment about apricots when ripe

A period would also be possible, and in casual writing many people might simply use a comma or two separate sentences. The semicolon just shows that the two parts are strongly connected in meaning.