Learn French: Le bulletin a annoncé que le week-end serait ensoleillé. - The bulletin announced that the weekend would be sunny.

This is because of the rules of reported speech. Since the main verb (a annoncé) is in the past tense, the future tense in the original quote must shift to the conditional. The original forecast was Le week-end sera ensoleillé (The weekend will be sunny), but when reported in the past, sera (will be) becomes serait (would be).

Does bulletin just mean a newsletter here?

In this context, le bulletin is typically short for le bulletin météo, which means the weather report or weather forecast. You will also see it used for school report cards (un bulletin scolaire) or news updates.

Is the word ensoleillé related to the French word for sun?

Yes! It is built around the word soleil (sun). By adding the prefix en- and the suffix -é, it becomes an adjective that means filled with sun, or sunny.

Can I leave out the word que like we often drop 'that' in English?

No, you cannot. In English, it is perfectly fine to say 'The bulletin announced the weekend would be sunny', but in French, the conjunction que is mandatory to connect the two clauses and can never be skipped.