Learn French: Le gouvernement affirme qu'il baissera les prix. - The government affirms that it will lower prices.

Because the introductory verb, affirme, is in the present tense. When you report speech using a present tense verb, you don't change the tense of the original quote. If the original statement was in the future tense, it stays in the future tense (baissera) in reported speech. Tenses only shift backward when the reporting verb is in the past.

Could we leave out qu' here, like we often drop the word that in English?

No, you can never drop que (or its shortened form qu' before a vowel) in French. While in English you can perfectly well say The government affirms it will lower prices, French strictly requires que to connect the reporting verb to the statement.

Why do we use il to refer to the government, instead of a plural pronoun?

In French, gouvernement is a singular masculine noun, so it must be replaced by the singular pronoun il. While English speakers sometimes treat collective nouns like government or team as a plural group, French strictly follows the grammatical singular.

I know baisser means to lower. Can it also mean to go down if prices drop by themselves?

Yes! The verb baisser is versatile. It works when someone lowers something (like the government lowering prices here) and also when something goes down on its own. If prices dropped naturally, you could just say Les prix baisseront (Prices will go down).