In addition to above solution,
you shall do this in a simpler way using hidden field
create a hidden field control in aspx
<asp:HiddenField ID="hdnfldList" runat="server" />
and you shall append the text from textbox to the literal control
protected void Button1_Click(object sender, EventArgs e)
{
ltrList.Text += TextBox1.Text + "<br/>";
TextBox1.Text = "";
}