Learn Spanish (Spain): Ayer cumplí treinta años.

In English, we "turn" an age, but in Spanish, you "complete" years. The verb cumplir means "to complete" or "to fulfill." When talking about having a birthday and reaching a new age, Spanish speakers always use cumplir.

I learned to use tener to talk about age. Why is it not used here?

You use the verb tener (to have) to state how old you currently are, like tengo treinta años (I am thirty). However, to talk about the specific event of having a birthday and becoming a new age, you must use cumplir.

Why is the verb cumplí in the preterite (simple past) tense?

The word ayer (yesterday) pinpoints a specific, finished event in the past. Because the action of having your birthday is completely finished, Spanish uses the preterite tense. Cumplí is the "I" form of this past tense.

Why isn't the word yo (I) included before the verb?

The ending of the verb already tells us who did the action. The -í ending on cumplí can only mean "I" did it. You could say Ayer yo cumplí treinta años for emphasis, but it sounds much more natural to drop the yo.