Hi  veenahosur,
I have created sample code which fullfill requirement.
HTML
<head runat="server">
    <title></title>
    <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
    <link href="http://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.0.3/css/bootstrap.min.css"
        rel="stylesheet" type="text/css" />
    <script type="text/javascript" src="http://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.0.3/js/bootstrap.min.js"></script>
    <link href="http://cdn.rawgit.com/davidstutz/bootstrap-multiselect/master/dist/css/bootstrap-multiselect.css"
        rel="stylesheet" type="text/css" />
    <script src="http://cdn.rawgit.com/davidstutz/bootstrap-multiselect/master/dist/js/bootstrap-multiselect.js"
        type="text/javascript"></script>
    <script type="text/javascript">
        $(function () {
            $('[id*=ddlCountries]').multiselect({
                includeSelectAllOption: true
            });
        });
    </script>
    <style type="text/css">
        body
        {
            font-family: Arial;
            font-size: 10pt;
        }
        table
        {
            border: 1px solid #ccc;
        }
        table th
        {
            background-color: #F7F7F7;
            color: #333;
            font-weight: bold;
        }
        table th, table td
        {
            padding: 5px;
            border-color: #ccc;
        }
    </style>
</head>
<body>
    <form id="form1" runat="server">
    <asp:GridView ID="GridView1" runat="server" AutoGenerateColumns="false" OnRowEditing="OnRowEditing">
        <Columns>
            <asp:TemplateField HeaderText="Name" ItemStyle-Width="150">
                <ItemTemplate>
                    <%# Eval("Name") %>
                </ItemTemplate>
                <EditItemTemplate>
                    <asp:TextBox ID="txtName" runat="server" Text='<%# Eval("Name") %>' />
                </EditItemTemplate>
            </asp:TemplateField>
            <asp:TemplateField HeaderText="Country" ItemStyle-Width="150">
                <ItemTemplate>
                    <%# Eval("Country") %>
                </ItemTemplate>
                <EditItemTemplate>
                    <asp:ListBox ID="ddlCountries" runat="server" SelectionMode="Multiple">
                        <asp:ListItem Text="United States" Value="United States" />
                        <asp:ListItem Text="India" Value="India" />
                        <asp:ListItem Text="France" Value="France" />
                        <asp:ListItem Text="Russia" Value="Russia" />
                    </asp:ListBox>
                </EditItemTemplate>
            </asp:TemplateField>
            <asp:TemplateField>
                <ItemTemplate>
                    <asp:LinkButton Text="Edit" runat="server" CommandName="Edit" />
                </ItemTemplate>
                <EditItemTemplate>
                    <asp:LinkButton Text="Update" runat="server" OnClick="OnUpdate" OnClientClick="return Validate(this)" />
                    <asp:LinkButton Text="Cancel" runat="server" OnClick="OnCancel" />
                </EditItemTemplate>
            </asp:TemplateField>
        </Columns>
    </asp:GridView>
    </form>
</body>
</html>
C#
protected void Page_Load(object sender, EventArgs e)
    {
        if (!this.IsPostBack)
        {
            DataTable dt = new DataTable();
            dt.Columns.AddRange(new DataColumn[3] { new DataColumn("Id"), new DataColumn("Name"), new DataColumn("Country") });
            dt.Rows.Add(1, "John Hammond", "United States");
            dt.Rows.Add(2, "Mudassar Khan", "India");
            dt.Rows.Add(3, "Suzanne Mathews", "France");
            dt.Rows.Add(4, "Robert Schidner", "Russia");
            ViewState["dt"] = dt;
            this.BindGrid();
        }
    }
    protected void BindGrid()
    {
        GridView1.DataSource = ViewState["dt"] as DataTable;
        GridView1.DataBind();
    }
    protected void OnRowEditing(object sender, GridViewEditEventArgs e)
    {
        GridView1.EditIndex = e.NewEditIndex;
        this.BindGrid();
        (GridView1.Rows[e.NewEditIndex].FindControl("ddlCountries") as ListBox).Items[e.NewEditIndex + 1].Selected = true;
    }
    protected void OnUpdate(object sender, EventArgs e)
    {
        GridViewRow row = (sender as LinkButton).NamingContainer as GridViewRow;
        string name = (row.FindControl("txtName") as TextBox).Text;
        string countryAdd = string.Empty;
        ListBox cblProject = (ListBox)(row.FindControl("ddlCountries"));
        foreach (ListItem item in cblProject.Items)
        {
            if (item.Selected)
            {
                countryAdd += item.Text + ",";
            }
        }
        countryAdd.Replace(",", " ");
        DataTable dt = ViewState["dt"] as DataTable;
        dt.Rows[row.RowIndex]["Name"] = name;
        dt.Rows[row.RowIndex]["Country"] = countryAdd;
        ViewState["dt"] = dt;
        GridView1.EditIndex = -1;
        this.BindGrid();
    }
    protected void OnCancel(object sender, EventArgs e)
    {
        GridView1.EditIndex = -1;
        this.BindGrid();
    }
Screenshot

I hope works for you.