Part of the reason you may have had trouble finding good resources on stateful firewalls is because, as Firewall.cx founder Chris Partsenidis mentions, most modern firewalls have a mix of abilities that place them in more than one firewall category.
Typically a stateful firewall associates some kind of state to each packet (IP-addresses, Port Numbers, L4-Protocol).
So it can associate a “connection” to every packet. (e.g. the well-known TCP-connection). See this article for a very good explanation. The difference between a stateful and stateless firewall, is that the stateless firewall only looks in the static information contained in the packet it is examining. As the stateful firewall memorizes a state-association to the packet, the filter can for example filter on the amount of data transferred for that TCP-connection.
If you need to understand the differences between the variety of firewalls out there, SearchNetworking’s Introduction to firewall types guide is another good resource.