Twilio developer evangelist here.

Once your account has alphanumeric sender IDs enabled, as described in the article you linked to, you can use them in one of two ways.

Either you set your alphanumeric sender ID as the From parameter when sending a message using the messages resource. This is explained in a bit more depth in this article about changing the sender ID for SMS messages or in this blog post I wrote about sending messages with alphanumeric sender IDs.

Or, you can create yourself a messaging service with Copilot and setup the alphanumeric sender ID there. This has the benefit of allowing you to setup a number pool to fall back on for sending to countries that don't support alphanumeric sender IDs, like the US.

From your Stack Overflow account it looks like you mainly program PHP for your back end. Here's a quick example:

use Twilio\Rest\Client;

// Your Account Sid and Auth Token from 
$sid    = "your_account_sid";
$token  = "your_auth_token";
$twilio = new Client($sid, $token);

$message = $twilio->messages
                  ->create($toNumber,
                           array(
                               "body" => "This is a branded message",
                               "from" => "ALPHA"
                           )
                  );

Let me know if that helps at all.