Listbox1.ToString() wont give you what you need..
try with
Listbox1.SelectedValue
private void btnDelete_Click(object sender, EventArgs e)
{
SQLiteConnection connection = new SQLiteConnection(connectionString);
SQLiteCommand delcmd = new SQLiteCommand();
delcmd.CommandText = "DELETE FROM Table WHERE ID='" + Listbox1.SelectedValue +"'";
connection.Open();
delcmd.Connection = connection;
delcmd.ExecuteNonQuery();
connection.Close();
bindListbox1();
Listbox1.Update();
MessageBox.Show("Row Deleted");
}