Leg Before Wicket (LBW) is never as straight forward as other forms of dismissals like Caught or Bowled. More often than not, there is that element of doubt as to whether
The ball would have gone to hit the wicket or
The ball hit in line with the wicket or
The ball pitched outside leg stump or
If there was an inside edge before the ball hit the stumps.
The full rules to give the batsman out LBW can be found here.
The decision making for the above can neither be given to the batsman nor the bowler as their decision could be biased towards their own team. Therefore, the only neutral person who would be in a position to judge without bias would be the umpire, and this is why the bowler/fielding team needs to appeal to get an LBW decision.
The umpire, however, cannot give a batsman out LBW, even if were to be plumb, if there is no appeal from the fielding team for the dismissal.(Law)