Learn Farsi: برای تولدت تبریک می‌گویم.

If you attached -at to the preposition, barây-at would mean "for you". But here, we specifically want to say "for your birthday", so the "your" ending (-at) must attach directly to the noun tavallod (birthday) to make tavallod-at.

The English translation says "on your birthday", but doesn't barâ-ye mean "for"?

Yes, barâ-ye translates directly to "for". Prepositions rarely match perfectly between languages. While English speakers congratulate someone "on" a special day, Farsi speakers literally say congratulations "for" it.

How does the verb tabrik mi-guy-am work here?

Farsi uses compound verbs very often. Instead of a single verb for "to congratulate", Farsi uses tabrik goftan, which literally means "to say congratulations". Here, goftan (to say) is conjugated in the present tense for "I" to become mi-guy-am.

Why isn't the direct object marker râ used after tavallod-at?

The word tavallod-at follows the preposition barâ-ye, making it the object of a preposition rather than a direct object. The marker râ is only used for specific direct objects, which are the main targets of the verb's action without a preposition in front of them.