A husband was complaining to his wife that she was spending too much time on her smartphone.

that...是 was complaining 的宾语从句。to his wife 是介词短语作状语：“对着妻子（抱怨）”。现在，把 to his wife 提到句首，就会一清二楚：

To his wife，a husband was complaining that she was spending too much time on her smartphone.

类似表达：

When I was nine or ten someone explained to me that when you are grown up you have to work 

我大概9岁或10岁的时候，有人跟我说人长大了就得工作。

《柯林斯高阶英汉双解学习词典》

She suggested to me that I might like to start regular savings and I said Well, I don't know whether I can stretch to that. 

她建议我可以开始定期存钱，我说我还不知道钱是不是够花呢。

《柯林斯高阶英汉双解学习词典》

He confided to me that he felt like he was being punished 

他向我诉说他觉得自己像是在受罚。

《柯林斯高阶英汉双解学习词典》

The fact that he had left her of his own accord proved to me that everything he'd said was true. 

他主动离开她这件事向我证明了他所言句句属实。

《柯林斯高阶英汉双解学习词典》

He indicated to me that we could never be good friends again. 

他向我暗示我们再也不可能成为好朋友了。

They grumbled to me that the decision was not fair. 

他们向我抱怨说那裁决不公平。

He whispered to me that there was an eavesdropper outside the window.

他悄声告诉我窗外有人在偷听。