Well, the long answer is that it's actually extremely complicated. Not really because the process is particularly difficult, but because the actual software implementations of these things run through so many layers these days. In principle, sending a text message works like this:

Both you and the recipient agree on a code, such that certain binary combination translate to certain letters. For instance, perhaps 00000100 means 'a'. How this code works is ultimately irrelevant, so long as both parties agree. And from there, it's as simple as taking a letter you put in, translate it to binary according to this code, send electrical signals that represent this binary code and then decode it on the other end to get out the letter again. You do that for every letter, and there you go: Text message. Essentially, it's just a fancier version of Morse code.

Now, how this actually works in reality is a lot more complex. In order to understand how sending text works, you need to know how the software you happen to use works. In order to understand that, you need to understand the framework it's built on. To understand that, you need to understand the operating system it runs on. And so on and so forth through understanding web sockets and protocols and the physical connections between servers and blablabla. But, ultimately, the basic idea behind it is as simple as encoding text in binary and decoding it on the other end.

I'm not sure if that really answers your question, as it still strikes me as very vague.

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!