verify the quote

There's zero evidence that this was ever said by Churchill, Truman, or anyone else involved in WWII. If one is to presume that the premise of the quote is accurate (that is to say, the war was fought by the allies to "conquer sales markets," why would they admit that after years of propaganda stating that the war was a war for freedom?

It's hard to source a negative, of course (aka, come up with a source saying that someone didn't say this particular line). The fact that there are no reputable sources that point towards Churchill or Truman actually uttering this line, however, should dispel its credibility.

explain the context in which it was made, and

Presumably by some Neo-Nazi, though I can't say who first created this statement. It fits in well with the Nazi idea that the Germans were the target of an international Jewish financier conspiracy.

also explain how historically true it was.

Completely false. It makes no sense - if Britain and France keen to "conquer" the sales market that was Germany, why would they follow a policy of appeasement throughout the 30s? Why wouldn't they have just conquered Germany in the 20s, when they were weak - say, just extend the occupation of the Ruhr to the entirety of the German nation?

The notion of one nation conquering another for economic gain is not absurd. After all, economic exploitation is one of the key features of imperialism. To portray Nazi Germany as a victim of imperialism, however, is absurd. The British and French were highly reluctant to go to war with Germany, but ended up doing so to prevent Germany from establishing hegemony over Europe. They couldn't have just pulled some strings and prevented Hitler from invading Poland without firing "one shot."

Your skepticism is well justified, and the fact that the quote only appears on site that are apologetic to the Nazis or based on attention-grabbing headlines is telling. It's not history, it's fiction.

Here's the text of Churchill's March 1946 speech at Fulton, called the "Iron curtain" speech because that was what popularized the phrase. Churchill doesn't even talk about the reasons for WWII, much less describing them as "to conquer sales markets." Instead, he focuses on the postwar situation, especially Anglo-American friendship and the Soviet threat.

As /u/EighthGear explains, that's not a statement Churchill would have said, both because it was inaccurate and because it would have contradicted his longtime position. But even beyond that, Churchill was a Nobel-prize-winning author and a longtime journalist who spoke with just pride of his grasp of English. He would've never used awkward phrases such as "conquer sales markets" and "without doing one shot."

Maybe the good people at /r/badhistory can help you ?

I agree with the other comments that the quote sounds like something that an apologist would create.