Hey nauna,

Please refer below sample.

HTML

<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<link rel="stylesheet" href="https://cdn.rawgit.com/dubrox/Multiple-Dates-Picker-for-jQuery-UI/master/jquery-ui.multidatespicker.css" />
<link rel="stylesheet" href="https://code.jquery.com/ui/1.12.1/themes/pepper-grinder/jquery-ui.css" />
<script type="text/javascript" src="https://code.jquery.com/ui/1.12.1/jquery-ui.min.js"></script>
<script type="text/javascript" src="https://cdn.rawgit.com/dubrox/Multiple-Dates-Picker-for-jQuery-UI/master/jquery-ui.multidatespicker.js"></script>
<script type="text/javascript">
    $(function () {
        $('[id*=txtMultiDatePicker]').multiDatesPicker({
            dateFormat: 'dd/M'
        });
    });
</script>
Date:
<asp:TextBox ID="txtMultiDatePicker" runat="server" />
<asp:Button ID="Button1" Text="Save" runat="server" OnClick="Save" />

Code

C#

protected void Save(object sender, EventArgs e)
{
    string Startdate = txtMultiDatePicker.Text.Split(',')[0];
    string Enddate = txtMultiDatePicker.Text.Split(',')[txtMultiDatePicker.Text.Split().Length - 1];
    ClientScript.RegisterClientScriptBlock(this.GetType(), "", "alert('Selected Dates are : " + Startdate + " to " + Enddate + "')", true);
}

VB.Net

Protected Sub Save(ByVal sender As Object, ByVal e As EventArgs)
    Dim Startdate As String = txtMultiDatePicker.Text.Split(","c)(0)
    Dim Enddate As String = txtMultiDatePicker.Text.Split(","c)(txtMultiDatePicker.Text.Split().Length - 1)
    ClientScript.RegisterClientScriptBlock(Me.GetType(), "", "alert('Selected Dates are : " & Startdate & " to " & Enddate & "')", True)
End Sub

Screenshot