Learn Turkish: Öğretmen gülümsedi çünkü anladı. - The teacher smiled because he understood.

This is due to 4-way vowel harmony. The past tense suffix changes its vowel to match the last vowel of the verb root. The root gülümse- ends in 'e', so it takes -di. The root anla- ends in 'a', so it takes -dı.

Where is the word for "he" in the second part of the sentence?

In Turkish, you don't always need to state the pronoun (like o for he/she/it) if it is clear from the context. Also, the 3rd-person singular past tense doesn't take an extra personal ending; the bare past tense suffix (-di or -dı) is enough to show it is a 3rd-person subject doing the action.

How do we know the teacher is a "he" from this sentence?

We actually don't! Turkish does not have gendered pronouns or gendered verb endings. Öğretmen simply means "teacher," and anladı can mean "he understood," "she understood," or "it understood." The English translation uses "he" here, but "she" would be completely correct as well.

I have seen the verb gülmek for laughing. Is gülümsedi related to that?

Yes, they are closely related! Gülmek means "to laugh," while gülümsemek specifically means "to smile." Adding the 3rd-person past tense suffix gives us gülümsedi (he/she smiled).