Learn Turkish: Güneş bugün bizim için doğdu.

This is because of 4-way vowel harmony. The past tense suffix is often -di, but since the only vowel in the verb root doğ- is an 'o', the suffix changes to match it, becoming -du. Because it is a third-person subject ('the sun'), there is no extra personal ending attached.

I thought doğmak meant 'to be born'? Why is it translated as 'rose'?

You are right! The literal and primary meaning of doğmak is 'to be born'. However, in Turkish, the sun doesn't just 'go up' in the morning; it is 'born'. Therefore, doğmak is the standard verb used for the sun rising.

The translation says 'for us', but doesn't bizim mean 'our'?

Yes, bizim means 'our'. In Turkish, when the postposition için (for) is used with a specific personal pronoun (like I, you, or we), that pronoun must take its possessive form. So instead of saying biz için, you must say bizim için (literally 'our for').

Why is bugün placed between the subject and the object?

Turkish is a Subject-Object-Verb language, which is why the verb doğdu is at the very end. Time words like bugün (today) typically sit near the beginning of the sentence, usually right after the subject (Güneş) or at the very start to set the scene for the action.