Why is 再 used in 家里的空调坏了，咱们另外再买一个吧?
另外 - In addition; other (adv)
再 - again (adv)
买 - buy (v)
Both 另外 and 再 are adverbs for the verb 买 in this sentence
Why both?
Because the sentence 另外买一个 could mean either 'buy an additional one' (now you have two air conditioners) or 'buy another one' (to replace the broken one)
If you say 再买一个, it could mean either 'buy an additional one' (now you have two air conditioners) or 'buy one again' (to replace the broken one)
Since the sentence already stated 家里的空调坏了, there's no reason to think you will have two air conditioners, therefore 另外买一个吧 would be good enough. Adding 再(again) avoids confusion and confirms the fact that the new one is a replacement. (we bought one before and now we have to buy one again - not the same one, another one)
&quot;家里的空调坏了，再买一个吧&quot;? is grammatical, but it would sound like you replace air conditioner often