What is meant by "Security Groups are stateful"?

It simply means if you allow say outbound rule on particular port protocol ip address it will allow to get back. So whatever coms in comes out.

You can compare this with NACL as they are stateless so you need to specify inbound and outbound rules in order to allow requested traffic flow.

I went to a movie with my son. During the film, he needed to go to the restroom.

The staff member at the door let him leave the theater and allowed him to re-enter afterwards. This is an example of a stateful request -- he went out and was allowed back in. However, somebody who simply tried to get into the theater would be rejected.

Similarly, your computer at home is connected to a router. The router protects it from the evils of the Internet. Traffic can't come through your router to your computer. However, if you make a request to go to a website, that request goes out of your router and the response is allowed back in because it is stateful. That is, the router remembers that you made the request to that website and it permits the response to come back to your computer.