"details why" is correct and usable in written English.
You can use it when you need to cite a reason for something. For example, "My report details why raising taxes on the wealthy would benefit the economy."
— Julia Moskin Details: Why the best bartenders are putting their concoctions in bottles.
– Glenn Collins Details: Why bartenders now want to lace our cocktails with acid phosphate.
A new study details why it's prudent to invest in carbon-free electricity now.
The paper, published in Nature Climate Change, details why some promises hold more water than others.
The presentation details why GM and other American auto manufacturers were not able to learn from NUMMI more quickly.
Myners dismisses his request for details: 'Why would we want to share commercial information with a competitor?' he asks.
When Lowell wrote about his marital griefs, he spared no details: "Why not say what happened?" he asks in a late poem.
WASHINGTON — My latest Letter From Washington details why Senator John Kerry of Massachusetts is an excellent choice to be secretary of state.
