These days, the companies who run phone networks (telecommunication carriers, aka "telcos" or "carriers") run all their networks via computers. If you send an SMS from one phone to another, it's likely already passing through multiple computers, so they can do things like collect logs for billing, queue messages for later delivery in case one phone is out of service, transfer messages to recipients on other carrier networks (eg. send it overseas via internet) etc.

These carriers might also provide a service that allows third parties to send SMS without using a phone, or connected to their cell network. Basically, one computer sends data to another computer run by the carrier - an SMS gateway - that will create the SMS message and push it out to the recipient phones. This generally costs money to use.

Some SMS gateways are used via web requests - like a REST API that many websites use today - others work by converting specific emails to SMS (eg. send an email to 123456789@sms.xyzcarrier.com, and the email body will be the text message).

There are also online services that act as a middleman, also confusingly calling themselves SMS Gateways. They deal with the various carriers across the world and their different gateways, figuring out which carrier to use based on the country and phone number, while providing a single unified service for end users and programmers to use with a simplified billing plan. This is likely what your cloud software is using on the server-side code.