Twilio evangelist here.
From the code above it does not look like you are replacing the {AccountSid}
token in the MessageApiString variable with your actual Account Sid.
Also, it looks like you are appending the phone number parameters to the URL as querystring values. Because this is a POST request I believe you need to include these as the request body, not in the querystring, which means you also need to set the ContentType property.
Here is an example:
var accountSid = "AC4*************0ab05bf";
var authToken = "0*************b";
var MessageApiString = string.Format("https://api.twilio.com/2010-04-01/Accounts/{0}/SMS/Messages.json", accountSid);
var request = WebRequest.Create(MessageApiString);
request.Method = "POST";
request.Credentials = new NetworkCredential(accountSid, authToken);
request.ContentType = "application/x-www-form-urlencoded";
var body = "From=+442033*****3&To=+447*****732&Body=test";
var data = System.Text.ASCIIEncoding.Default.GetBytes(body);
using (Stream s = request.GetRequestStream())
{
s.Write(data, 0, data.Length);
}
var result = request.GetResponse();
Hope that helps.