Let's take this step-by-step to see how the Heat broke down the Celtics defense to get Cole such an open look. Boston opens the possession in something in between a man-to-man defense and a 3-2 zone. Through the six seconds of the play, they appear to be following zone principals, so we'll call it that.
The Heat open the play with a LeBron James/Chris Bosh high pick-and-roll. Pick-and-roll plays don't always work well against a zone (and one of the primary reasons teams go to a zone against Miami is to minimize their ability to run P-N-R with two of their big three), but Miami is using the P-N-R here to set up secondary actions.
As LeBron comes around the Bosh pick (a strong pick that seals Keyon Dooling off from the rest of the play), Dwyane Wade crashes the lane and -- this is important -- Norris Cole fills the vacated spot.
Here, as James turns the corner, you can see how Boston's zone is already starting to break down. Rajon Rondo lets Wade go into Allen's zone, as he should, but also collapses on James, which makes sense, as LeBron is the Heat's most dangerous player in this situation, and the goal should be to force the ball out of his hands.
With Rondo crashing down, LeBron jumps to pass (NO!) and has three options at this point, since Bosh turned the pick-and-roll into a basic ball screen. From left to right, option one is Udonis Haslem in the corner, but Brandon Bass has crept into the passing lane, cutting that off. Option 2 is a dangerous pass in the lane to Wade, one that would likely be deflected or intercepted by Kevin Garnett. Even if it wasn't, Wade has Allen at the rim ready to stop him, and a charge is a likely possibility in this situation. Option 3 is Cole, who is making his way from the corner to the wing, putting Wade between him and Allen.
By the time Cole catches the ball and sets to shoot, he's still got considerable distance between himself and Allen. If he'd stayed in the corner -- like James Jones or Mike Miller would have -- James has a longer pass to make and Allen has more time to recover, and less distance to cover on that recovery. As it is, Cole gets a look that most NBA guards should be able to make with reasonable regularity, a look he doesn't get if LeBron doesn't draw Rondo's attention.
It's also worth noting that because Bass has to respect Haslem's jumper, he can't be counted on to cover the Wade crash, which holds Allen in place. As far as Boston's defense is concern, the Cole jumper is the lesser of all evils, but letting him get that wide open wasn't in Boston's plan.