Hi Sid,
I have created one sample that full-fill your requirement.
HTML
<div>
    <asp:Label ID="lblConnection" runat="server" /><br />
    <asp:Button ID="btnConnection" Text="Read ConnectionString" runat="server" OnClick="btnConnection_Click" />
</div>
Code
protected void btnConnection_Click(object sender, EventArgs e)
{
    string txtpath = @"D:/Connection/ConnectionString.txt";
    StreamReader sr = new StreamReader(txtpath);
    String line = sr.ReadToEnd();
    lblConnection.Text = line;
    //use this in your code.
}
Screenshot
 
But why are you storing connection string in a text file instead of app.config/web.config. Which is more secure and manageable.