Rugby presents a unique challenge in that the referee is required to make a specific decision about a contested tackle almost 200 times a match. That is one every 30 seconds roughly and in a game he has to make over a 100 decision in a split second.
Different referees have a different sequence or approach to the decision, but they must judge, more or less in order: how the tackler interacts with the tackled player, when the tackle actually occurs, that the tackler releases the tackled player, that the tackled player releases the ball, when the ruck is formed, that players arriving to join the ruck remain on their feet, and that they join from the correct position and do not seal the ball off to prevent the contest. Add in that there are often multiple tacklers, so the referee has to decide who the tackler is, and you appreciate that within half a second, there's a lot to judge. Then the next problem is that many times, four or five things happen more or less simultaneously, and so it really is a judgment call.
Ultimately, what the decision comes down to is a) assigning roles to the involved players, and b) deciding on the order in which events occur every tackle has similar events, and the job of the referee is to sort through the order in which they occur, and if he sees a different order to you or I, then his decision will be accordingly different. And this is precisely what happens to make these decisions so contentious.
