I have no doubt that Labour screwed up their part of the Remain campaign. They were barely audible to the UK public and their leader was useless at persuading wavering Labour voters. He left Ed "loser boy" Milliband to do most of it. They even had to bring on Gordon Brown at the last minute to try to turn things around! (He was a hopeless former Labour leader and one of the worst PMs ever.)
I suspect what happened was that the lefty voters in Labour constituencies voted for Leave in order to give David Cameron and George Osborn (the posh Conservative boys) bloody noses. Cameron and Osborn were the most effective Remain campaigners but the fact that they are Conservatives did not endear them to disgruntled Labour voters. The Labour Party dropped the ball big time, but they have always been experts at screwing up!
sounds familiar ...