ODO's entry for slide defines "let something slide" as:
negligently allow something to deteriorate:
Papa had let the business slide after Mama’s death
Therefore, the connotation is quite literal in that you are letting something slide down. Furthermore, there is also an implicit restriction involved in your expression which would expand to something along the lines of:
Let it slide this time
In other words, you're allowing the situation to deteriorate temporarily.
The etymology of the word slide dates the first use of let (something) slide to the 14th century:
O.E. slidan "move smoothly, glide," from P.Gmc. *slidanan (cf. O.H.G. slito, Ger. Schlitten "sledge"), from PIE root *(s)lei- "slide" (cf. Lith. slystu "to glide, slide," O.C.S. sledu "track," Gk. olisthos "slipperiness," olisthanein "to slip," M.Ir. sloet "slide"). Phrase to let (something) slide is recorded from late 14c.
I think the key to understanding that phrase you quoted is another quote from the article:
In other developments, available phone numbers ran out, forcing the introduction of unpleasant new area codes. “Awesome” went from being a risible word used only by stoners and surfers to an acceptably ubiquitous modifier, the Starbucks of adjectives.
Twenty years ago, someone saying: "Awesome!" (or also commonly: "Totally awesome, dude!") was, as the author says, something from the surfer/stoner subculture. It would not get much mileage in mainstream use.
Nowadays, particularly among the younger generation, it is very commonly used. You can substitute "Great!" or "Excellent!" and get the same meaning.
However, as the other answers suggest, it can also be used disingenuously, as in "Oh, that's nice." That's why the author feels the need to clarify "And I think he meant it..." to show that the clerk was really saying it was great.
What was the clerk saying was great? It could be open to interpretation. Given that the author uses that line to close the article, which is about moving back to New York, I would say they meant it to refer to the move in general.
Best Answer
It means the same as "What am I alleged to have stolen?", where the emphasis is deliberately placed to make the listener aware that one is not conceding the point that one has, in fact, stolen anything.