You're not executing any show() method in the watchlist click event, this is probably why nothing is showing. To fix it just replace the event with:
$("#watchlist").click(function(){
$(".completed").hide(100);
$(".fav").hide(100);
$(".watchlist").show(100);
});
The way you created this will result in an exponentially large script when you're planning to add more things in the future. I would advise to generalize it a little bit.
HTML:
<a href='#' rel='anime' class='toggler'>All</a>
<a href='#' rel='fav' class='toggler'>Favorites</a>
<a href='#' rel='watchlist' class='toggler'>Watchlist</a>
<a href='#' rel='completed' class='toggler'>Completed</a>
Javascript:
$(".toggler").click(function(e) {
e.preventDefault();
$(".anime").hide(100);
$("." + $(this).attr('rel')).show(100);
});
Since the show comes after the hide, you don't have to use :not() in your CSS selectors. You could create an if statement for the all button so that not everything hides before showing.