To understand this feature, imagine the chat as a rope that enables two people to talk to each other. Whoever has one end will be able to converse with the person on the other end of the rope. When a visitor on a website starts a chat, they have one end of this rope tied to their browser, and the other end is tied to Apexchat's chat software where an agent is handling the chat. The agent and the visitor will continue chatting to each other until one of them either let go of their end of the rope or hand it over to another person. The transfer feature is basically one person handing off to their end of the rope to another person or tieing one end of the rope to another chat medium (web browser, SMS, call, etc).
In web transfer, the chat starts with the visitor's browser on one end and the chat agent with Apexchat software on another end. When transfer happens the chat agent hand over the chat to the client's own chat operator, in other words, the chat agent hand over their end of the rope to the chat operator on the mobile app or the web application. This way the chat operator continues the rest of the chat with the visitor.
With SMS transfer the chat starts exactly the way it would start for webs transfer but when it is transferred the agent hands off their end to the operator on the ApexConnect mobile app and the visitor's end is disconnected from the browser and the visitor receive an SMS on their mobile number that lets them continue the same chat on SMS when they respond back to it. So SMS transfer doesn't just transfer chat from Apexchat agent to client's own operator but also moves visitor from web browser to SMS.