That narration is wrong with the tenses. Our neighbour downstairs, who WAS a man of more than fifty years , WOULD often KNOCK at our door and asks me, "Did you hear that noise a short while ago?" I would then think back to just a few moments before and I would be sure that we hadn't made any noise at all, and I would patiently explain this to him. The change of tense in the original is a little confusing. I don't know why it was done that way.

'Would' is used for two things: 1) As a form of 'will' for hypothetical situations. 2) A past tense form of 'will. You could argue that those could fall under one definition, but it's easier to consider them as two. 'Would'有两个用途: 第一：我不知道怎么说明它是因为我汉语说得不好。'Would‘是'will', 但是'would'跟假设的情况(hypothetical situations?)被用。 第二：能是"will"过去时态的形态(form?). 可能这些真的只是一个用途，但是如果人以为它们是两个，就可能更容易明白。 在你的句子中，'would‘表达等东西发生，别的东西再发生。句子是过去时态，所以'will’成'would'。

Another way to write this would be as follows: I thought back to just a few moments before and I was sure that we hadn't made any noise at all, and I patiently explained this to him. Both sentences are in past tense, but the original has a little more meaning. By saying 'would', it emphasizes his actions, like saying, "I would have to tell the old man again and again that it wasn't me that made the noise, but he kept coming back." He said the the man came often. When I rewrote the sentence without the 'would', it sounds like the first time he told the mad that he didn't make the noise. The original sentence with the 'would ' implies that he's had to explain this to the man several times.

"would" is used as a part of unreal situation open your grammar book at "Conditional Mood"