Edit in JSFiddle

 var treeView = new wijmo.nav.TreeView('#TreeView', {
     itemsSource: createItem(),
     displayMemberPath: 'header',
     childItemsPath: 'items',
     showCheckboxes: true
 });

 var st = '';
 function btn_click() {
     var treeNodes = treeView.nodes;

     st = '';
     for (var i = 0; i < treeNodes.length; i++) {
         getCheckItems(treeNodes[i]);
     }

     alert(st);
 }

 function getCheckItems(node) {
     if (node.isChecked) {
         st = st + node.dataItem.header + ', ';
     }
     if (node.hasChildren) {
         for (var i = 0; i < node.nodes.length; i++) {
             getCheckItems(node.nodes[i]);
         }
     }
 }
 
 function createItem() {
     var items = [
         {
             header: 'Electronics', items: [
               { header: 'Trimmers/Shavers' },
               { header: 'Tablets' },
               {
                   header: 'Phones', items: [
                     { header: 'Apple' },
                     { header: 'Motorola' },
                     { header: 'Nokia' },
                     { header: 'Samsung' }]
               },
               { header: 'Speakers' },
               { header: 'Monitors' }]
         },
         {
             header: 'Toys', items: [
               { header: 'Shopkins' },
               { header: 'Train Sets' },
               { header: 'Science Kit' },
               { header: 'Play-Doh' },
               { header: 'Crayola' }]
         },
         {
             header: 'Home', items: [
               { header: 'Coffeee Maker' },
               { header: 'Breadmaker' },
               { header: 'Solar Panel' },
               { header: 'Work Table' },
               { header: 'Propane Grill' }]
         }
     ];

     return items;
 }

<input type="button" onclick="btn_click()" value="チェックされた項目" />
<div id="TreeView"></div>