The company offered little 'by way of explanation' for the delays. [=as a way of explaining why there were delays]
Some of the more technical terms need/require 'explanation'. [=they need to be explained]
a likely/probable/possible/plausible 'explanation of' how the accident occurred
(..) All right? I will explain the whole thing to them. And while you're at it, explain it to me. I don't owe you an explanation. I owe you an apology, and I have apologised. I am very sorry, Gretchen. There. I've apologised twice now. I'm humbly sorry. (..)2009 TvShow: Breaking Bad Title: Peekaboo Season: 2 Episode 6