Коли мені виповнилося десять років, я отримав перший телефон.

Here виповнилося means turned or reached the age of.

So мені виповнилося десять років means I turned ten years old.
The verb comes from виповнитися, and in age expressions it is a very common way to talk about birthdays or reaching a certain age.

Because this Ukrainian age expression uses the dative case for the person.

So Ukrainian says:

мені виповнилося десять років = literally something like to me, ten years were completed

That sounds strange in English, but it is normal in Ukrainian.
So мені is correct, and я виповнилося... would be wrong.

This is a standard impersonal construction in Ukrainian.

In this pattern, the past-tense verb usually appears in the neuter singular form: -лося. That is why you get:

• мені виповнилося десять років
• йому виповнилося двадцять років
• їй виповнилося п’ятнадцять років

Even though the age is ten years, the verb does not become plural here.

Because after numbers 5 and higher, Ukrainian normally uses the noun in the genitive plural.

The forms with рік are:

• 1 рік
• 2, 3, 4 роки
• 5+ років

So десять років is exactly what you expect.

They are similar, but not identical.

• Коли мені виповнилося десять років = When I turned ten
• Коли мені було десять років = When I was ten

The sentence you have focuses on the specific moment of turning ten.
The version with було talks more generally about the time when you were ten.

Because отримав is perfective, and it describes a single completed event: you got the phone.

• отримав = got / received
• отримував = was receiving / used to receive

In this sentence, getting the first phone happened once, so отримав is the natural choice.

Because in Ukrainian, the singular past tense shows the speaker’s gender.

• я отримав = said by a male speaker
• я отримала = said by a female speaker

So this sentence sounds as if the speaker is male.

It is in a different case: it is accusative. But for masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: перший телефон
• accusative: перший телефон

That is why the form does not visibly change.
The adjective перший also stays the same here for the same reason.

In this context, it means my first phone.

Ukrainian often leaves out possessives like мій when the meaning is already obvious from context. Since the speaker is talking about what they received, перший телефон naturally means their first phone.

If you said мій перший телефон, that would add emphasis, but it is not necessary.

Because Коли мені виповнилося десять років is a subordinate clause introduced by коли.

Ukrainian normally separates this kind of clause from the main clause with a comma:

Коли мені виповнилося десять років, я отримав перший телефон.

This is similar to English: When I turned ten, I got my first phone.

Yes. Ukrainian word order is fairly flexible.

You could also say:

Я отримав перший телефон, коли мені виповнилося десять років.

That means the same thing.
The original version puts the time clause first, which gives a little more emphasis to when it happened.