score:0

Accepted answer

You run 53 times your animate function for only one $('.quick-contact')

I have add a variable active = 1 for check is animated.

Please try, it's work fine

$(document).ready(function() {
    var active = 1;
    /* Every time the window is scrolled ... */
    $(window).scroll( function(){
    
        /* Check the location of each desired element */
        $('.quick-contact').each( function(i){
            
            var bottom_of_object = $(this).offset().top + $(this).outerHeight();
            var bottom_of_window = $(window).scrollTop() + $(window).height();
            
            /* If the object is completely visible in the window, fade it it */
            if( bottom_of_window > bottom_of_object && active ==1){
                $(this).delay( 1000 ).animate({'opacity':'1'},500);
                $(this).animate({'bottom':'0px'},500);
                active = 0;
            }
            
        }); 
    
    });
    
});
$(document).ready(function(c) {
	$('.alert-close').on('click', function(c){
		$('.quick-contact').fadeOut('fast', function(c){
	  		$('.quick-contact').remove();
		});
	});	
});
.wrapper
{
  width:100%;
  height:1500px;
  position:relative;
}
.quick-contact
{
  width:300px;
  height:200px;
  position:fixed;
  bottom:200px;
  right:0px;
  padding:0px;
  background:#f6f6f6;
  color:#333333;
  opacity:0;
  z-index:99;
}
.alert-close
{
  color:red;
  cursor:pointer;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="wrapper">
<div class="quick-contact">Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua.<div class="alert-close"><div class="close-btn">Close me</div>
 </div>
</div>
</div>

codepen : http://codepen.io/anon/pen/YwOedm

UPDATE

$(document).ready(function() {

    $(window).scroll( function(){
      if($(window).scrollTop() + $(window).height() == $(document).height()) {
        $(".quick-contact").delay( 1000 ).animate({'opacity':'1'},500);
        $(".quick-contact").animate({'bottom':'0px'},500);
      }
    });
    
});
$(document).ready(function(c) {
	$('.alert-close').on('click', function(c){
		$('.quick-contact').fadeOut('fast', function(c){
	  		$('.quick-contact').remove();
		});
	});	
});
.wrapper
{
  width:100%;
  height:1500px;
  position:relative;
}
.quick-contact
{
  width:300px;
  height:200px;
  position:fixed;
  bottom:200px;
  right:0px;
  padding:0px;
  background:#f6f6f6;
  color:#333333;
  opacity:0;
  z-index:99;
}
.alert-close
{
  color:red;
  cursor:pointer;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="wrapper">
<div class="quick-contact">Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua.<div class="alert-close"><div class="close-btn">Close me</div>
 </div>
</div>
</div>


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