JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/2.0.4/css/bootstrap-responsive.css">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/2.0.4/css/bootstrap.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/1.7.1/jquery.min.js"></script>
<div class="input-group control-group after-add-more">
  <input type="text" name="addmore[]" class="form-control" placeholder="Enter Name Here">
  <div class="input-group-btn">
    <button class="btn btn-success add-more" type="button"><i class="glyphicon glyphicon-plus"></i> Add</button>
  </div>
</div>
<div class="copy-fields hide">
  <div class="control-group input-group" style="margin-top:10px">
    <input type="text" name="addmore[]" class="form-control" placeholder="Enter Name Here">
    <div class="input-group-btn">
      <button class="btn btn-danger remove" type="button"><i class="glyphicon glyphicon-remove"></i> Remove</button>
    </div>
  </div>
</div>

JavaScript

$(document).ready(function() {
   $(".add-more").click(function() {
     var html = $(".copy-fields").html();
     $(".after-add-more").after(html);
   });
   $("body").on("click", ".remove", function() {
     $(this).parents(".control-group").remove();
   });
 });