This is a great piece of word-painting. The speaker was Major-General John Sedgwick, a main figure in the American Civil war. The whole way to understand florid sentences like this is to apply the phrase "as if it were..." Simply, Sedgwick gave an inspiring speech. However, it wasn't any speech; it was as if it were a sermon, full of religious importance. And not just any sermon; it was as if this sermon were a labourer working furiously, covered in sweat. So there's your "sweating sermon". What's more, Sedwick simply didn't deliver a "sweating sermon"; he spoke so passionately it was if he were thrashing the sermon like a beast of burden. "He put off his doublet" sounds Elizabethan. No-one wore doublets in Sedgwick's time. This finish to the description of Sedgwick's speech makes the reader imagine that not only was Sedgwick passionate (thrashing a sweating sermon, and so on), but also highly poetic, as if he had just declaimed Shakespeare before The Bard himself. In layman's terms, Sedgwick needed to take of his coat and take a rest. Maybe. What an image!! Just use your imagination. :)

This sentence is constructed properly, but it makes no sense. To thrash is to whip or beat -- how do you beat a sermon? How does a sermon sweat? Wouldn't the thrasher sweat (or the listener, if the sermon is frightening)? What does a doublet have to do with any of this? It's nonsense.

Since I know it's from 17th Century English, it makes a bit more sense. We don't use "thrash" that way anymore, but it means he delivered the sermon forcefully. We would not use the word "sweating," but we might speak of a "blistering" sermon, which means full of fire and brimstone or strong moral judgements. So the preacher delivered a blistering sermon so forcefully that it caused him to take off his coat (to cool off).