Illegal use of ngTransclude directive in the template

The reason is when the DOM is finished loading, angular will traverse though the DOM and transform all directives into its template before calling the compile and link function.

It means that when you call $compile(clone.children()[0])(scope), the clone.children()[0] which is your <panel> in this case is already transformed by angular.
clone.children() already becomes:

<div ng-transclude="">fsafsafasdf</div>

(the panel element has been removed and replaced).

It’s the same with you’re compiling a normal div with ng-transclude. When you compile a normal div with ng-transclude, angular throws exception as it says in the docs:

This error often occurs when you have forgotten to set transclude:
true in some directive definition, and then used ngTransclude in the
directive’s template.

DEMO (check console to see output)

Even when you set replace:false to retain your <panel>, sometimes you will see the transformed element like this:

<panel class="ng-scope"><div ng-transclude=""><div ng-transclude="" class="ng-scope"><div ng-transclude="" class="ng-scope">fsafsafasdf</div></div></div></panel>

which is also problematic because the ng-transclude is duplicated

DEMO

To avoid conflicting with angular compilation process, I recommend setting the inner html of <panel1> as template or templateUrl property

Your HTML:

<div data-ng-app="app">
        <panel1>

        </panel1>
    </div>

Your JS:

app.directive('panel1', function ($compile) {
            return {
                restrict: "E",
                template:"<panel><input type="text" ng-model="firstName">{{firstName}}</panel>",

            }
        });

As you can see, this code is cleaner as we don’t need to deal with transcluding the element manually.

DEMO

Updated with a solution to add elements dynamically without using template or templateUrl:

app.directive('panel1', function ($compile) {
            return {
                restrict: "E",
                template:"<div></div>",
                link : function(scope,element){
                    var html = "<panel><input type="text" ng-model="firstName">{{firstName}}</panel>";
                    element.append(html);
                    $compile(element.contents())(scope);
                }
            }
        });

DEMO

If you want to put it on html page, ensure do not compile it again:

DEMO

If you need to add a div per each children. Just use the out-of the box ng-transclude.

app.directive('panel1', function ($compile) {
            return {
                restrict: "E",
                replace:true,
                transclude: true,
                template:"<div><div ng-transclude></div></div>" //you could adjust your template to add more nesting divs or remove 
            }
        });

DEMO (you may need to adjust the template to your needs, remove div or add more divs)

Solution based on OP’s updated question:

app.directive('panel1', function ($compile) {
            return {
                restrict: "E",
                replace:true,
                transclude: true,
                template:"<div ng-transclude></div>",
                link: function (scope, elem, attrs) {
                    elem.children().wrap("<div>"); //Don't need to use compile here.
                   //Just wrap the children in a div, you could adjust this logic to add class to div depending on your children
                }
            }
        });

DEMO