Use the Selected index changed event for first dropdownlist and set autopostback=true for first dropdown
protected void dropdownMonth_SelectedIndexChanged(object sender, EventArgs e)
{
getDays(dropdownMonth.SelectedValue);
}
set AutoPostBack="true" for dropdownDays also,
protected void dropdownDays_SelectedIndexChanged(object sender, EventArgs e)
{
LabelDays.Text = dropdownDays.SelectedValue;
}