You have some mistakes in javascript codes.
First, you have to show/hide textbox only for that section. In order to do this, you can use closest() and find() methods of jQuery.
Secondly, when checking checked property, it is better to use $(this) of jQuery because javascript's this is something different there.
Finally, when clonning an element you should use true parameter if you want to clone actions of elements, too.
Working codes are:
$( document ).ready(function() {
$("#tijdstiptoevoegen").on( "click", function(e){
e.preventDefault();
$( ".tijdstip" ).clone(true).appendTo( ".tijdstippen" );
});
$(".chkUitleg").on("change",function() {
if($(this).is(":checked")) {
$(this).closest(".tijdstip").find(".verdere_uitleg").show();
}
else
{
$(this).closest(".tijdstip").find(".verdere_uitleg").hide();
}
});
});