Remember UpstairsDownstairs? Mr Hudson was always saying "Look sharp!" to Ruby and the others. It was translated (here in Sweden) as "And be quick about it!" What's the origin; sounds like something a really bad teacher might say to a pupil, meaning something like "could you at least look attentive" ...? But how does that relate to being quick...?
I think it originally meant, 'be alert' or 'be extra vigilant'. But over time it changed to mean something like 'move as though the matter at hand is urgent'.
'...Be quick about it...' is a good translation.
As Leo suggests, the phrase originally meant to look sharply after something...ie keep a strict watch on it. Later it took on the colloquial meaning of move oneself briskly or lose no time about whatever one was doing. It had the former meaning from the early 18th century and took on the latter about a century later. (The former is now no longer in use.)
Oh I see! Almost forty years of wondering have now come to an end ;-) I was completely stuck in equating "look" with "appear to be", and that didn't make sense did it. Thanks everyone.