What this comes down to is positioning, I think. Watch carefully - I think the AR got caught ball watching, and it went downhill from there. Now, don't get me wrong - even being out of position the AR should have gotten this one right, and the referee should have been able to overrule the AR here. But watch the AR carefully. As the ball is moving out from goal (before the shot), the AR seems to get caught ball watching a little and is a bit behind the 2nd last line of defence. He realises this and takes some very rapid side steps to catch up. What this means is that as the shot is taken the AR is moving UPFIELD, pulling him out of a spot where he can tell if it's clearly crossed the line. From there we can only assume it's a trick of the angle or the keeper blocking his view of the ball hitting the net, perhaps compounded by the AR's main priority at this moment is getting back into position, so that's occupying his mind. But basically, the AR's mind is probably focussed on catching up and the offside line, meaning he isn't able to dedicate enough mental processing power to the incident at hand. If he was in correct position, then the moment the ball is kicked he doesn't need to think too much about offside position, so 'ball over the line' becomes his primary thought as he sprints to the line.
What SHOULD have happened is the AR is right in line with the 2nd last defender and as soon as the shot is taken, start sprinting for the line (situational awareness is so important). He would have been much closer to the line then and been able to focus on the ball.
Now, the ball hit the side net, not the back. So I think the AR causing a problem has made the referee doubt himself, when the ref would no doubt have given it if he was by himself.
