They'll usually use a service like Amazon Web Services Simple Notification System, or Twilio. These allow you to send text messages using a web request just like how you load a web page. Then it is just a case of writing a piece of software to decide when to send the request and what to put into it.

Those services I mentioned will send the text message from a number or alphanumeric sender ID (where you see text messages from a sender who is a word or phrase rather than a phone number) into the network of the carrier that the recipients SIM card is registered with.

Twilio will even notify your software if the message gets rejected because it is considered spam, or the carrier doesn't support receiving messages from Twilio, or if they have their phone switched off and the message doesn't go through. It is pretty cool how it works and how easy it is to use!