Here's a slightly modified version of your code with ng-show and animate.css working together.
HTML
<div ng-controller="controller">
  <button ng-click="showme = !showme">Show or Hide</button>
  <div ng-show="showme" class="boxme">
    <h2>Box to Show and Hide</h2>
    <p>Show and hide this box using animate.css the angularway!</p>
  </div>
</div>
JavaScript
(function() {
    var app = angular.module("theapp", ['ngAnimate']);
    var controller = function($scope){
        $scope.showme = false;
    };
    app.controller("controller", controller);
}());
CSS
.boxme.ng-hide-add {
    -webkit-animation: fadeOutLeftBig 0.4s;
    display: block!important;
}
.boxme.ng-hide-remove {
    -webkit-animation: fadeInLeftBig 0.4s;
}
http://jsfiddle.net/h58wsxcw/
Note the options on the left (body tag, external resources) that have to be set to get angular up and running on jsfiddle.