Части даты можно получить из атрибута даты DatePicker в функции onSelect:
$("#datepicker").hide();
$("#buttonHere").click(function() {
$("#datepicker").toggle();
});
$("#datepicker").datepicker({
onSelect: function(value, date) {
const months = ["January", "February", "March", "April", "May", "June",
"July", "August", "September", "October", "November", "December"
];
$('#resultdate_expectation span:eq(0)').html(months[date.selectedMonth]);
$('#resultdate_expectation span:eq(1)').html(date.selectedDay);
$('#resultdate_expectation span:eq(2)').html(date.selectedYear);
$("#datepicker").hide();
}
});
jsFiddle, пример