Learn Turkish: Onun yaşı kaç?

The base word is yaş, which means 'age'. Because we are talking about 'his' age, we need to add the 3rd person singular possessive suffix. Since yaş ends in a consonant and its last vowel is an 'a', the 4-way vowel harmony rule tells us to add -ı, making it yaşı.

The lesson mentions a buffer 's' for this suffix, so why isn't it yaşsı?

The buffer 's' is only used when the base word ends in a vowel. This prevents two vowels from crashing together (like in arabası for 'his car'). Because yaş already ends in a consonant, we do not need the buffer letter and can attach the vowel suffix directly.

Could I drop onun and just ask 'Yaşı kaç?'

Yes! Because the suffix -ı on yaşı already tells us we are talking about his, her, or its age, the word onun is structurally optional. You usually include it only for emphasis, or if it isn't clear from the context who you are talking about.

Why do we use kaç (how many) instead of a word for 'what'?

In Turkish, age is treated as a number quantity that someone has. So instead of asking 'what' their age is, you are literally asking 'how many' their age is by using the question word kaç.