Learn Turkish: Ben alfabeyi üç yaşında öğrenmişim. - Apparently I learned the alphabet at the age of three.

The word öğrenmişim uses the -miş past tense, often called the reported or evidential past. Turkish uses this tense for things we did not experience directly, or things we do not personally remember. Since the speaker was only three years old, they do not have a clear memory of learning the alphabet and are reporting a fact they likely heard from their parents.

How does üç yaşında translate to "at the age of three"?

It literally translates to "at its age of three." It is built using the number üç (three) plus yaş (age), the possessive suffix -ı, the buffer letter -n-, and the locative suffix -da (in/at). You can use this exact pattern for any age, such as beş yaşında for "at the age of five."

Why does alfabe have the -yi ending?

The base word is alfabe (alphabet). Because it is the specific, known thing being learned, it acts as a direct object and needs the definite accusative case suffix. Since alfabe ends in a vowel, we add the buffer letter -y- before the accusative suffix -i, making it alfabeyi.

Is the word Ben strictly necessary at the beginning?

No, you could completely drop it. The personal suffix -im at the very end of öğrenmişim already tells us that "I" am the person who learned it. Including Ben just adds a little bit of emphasis to the sentence.