You can not echo directly nor redirect inside ajax call. What you need is to echo or redirect using javascript/jquery inside ajax callback

So e.g. in controller you have

public function destroy($id) {       
       $category = Category::find($id);
       $category->delete();   
       return 'success'; //not the best value to return, but for illustration :)
    }

Then in jquery. This will alert the 'success' string from controller

$.ajax({
     type: "DELETE",
     url: "{{ URL::to('admin/categories/" +idtoedit +"')}}",
     success: function (result) {
        alert(result);  
     }
});

If you want to redirect, you must do it in jquery too

$.ajax({
    type: "DELETE",
    url: "{{ URL::to('admin/categories/" +idtoedit +"')}}",
    success: function (result) {
         window.location.replace('/admin/categories');  
    }
});