jQuery Simple Example

HTML

<script src="//www.addressfinder.co.nz/assets/v2/widget.js"></script>
<div class="autogrow" contenteditable="true"></div>

CSS

.autogrow {
  border: 1px solid rgb( 0, 0, 0 );
  padding: 10px;
}
@keyframes line_insert {
  from {
    height: 0px;
  }
  to {
    height: 20px;
  }
}
.autogrow[contenteditable] > div {
  animation-duration: 250ms;
  animation-name: line_insert;
  //height:20px;
}
.autogrow[contenteditable] {
  overflow: hidden;
  line-height: 20px;
}

JavaScript

$(".autogrow").keydown(function(e){

    if (e.keyCode == 13 && e.shiftKey || e.keyCode == 13)
    {
       $(this).animate({height: $(this).height()+20},200);
				$(this).append('<div><br></div>');
    }
});